💡Okay, so I need to find the derivative of the function f(x) = sin(x)/x. Hmm, I remember that when you have a function like this, which is a quotient of two functions, you can use the quotient rule to find its derivative. Let me recall what the quotient rule is. I think the quotient rule says that if you have a function f(x) = u(x)/v(x), then the derivative f'(x) is equal to [u'(x)v(x) - u(x)v'(x)] divided by [v(x)] squared. So, in mathematical terms, f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]². Yeah, that sounds right.Alright, so in this case, my function is f(x) = sin(x)/x. Let me identify u(x) and v(x). Here, u(x) is sin(x) and v(x) is x. Got it. Now, I need to find the derivatives of u(x) and v(x). The derivative of u(x) = sin(x) is u'(x) = cos(x). That's straightforward. And the derivative of v(x) = x is v'(x) = 1. Simple enough.Now, plugging these into the quotient rule formula. So, f'(x) should be [u'(x)v(x) - u(x)v'(x)] divided by [v(x)]². Substituting the values, that becomes [cos(x) * x - sin(x) * 1] divided by x squared. Let me write that out: f'(x) = [x cos(x) - sin(x)] / x². Hmm, that seems correct. Let me double-check to make sure I didn't mix up any signs or terms. Wait, the quotient rule is [u'v - uv'] / v², right? So, u' is cos(x), v is x, so the first term is cos(x)*x. Then, u is sin(x), v' is 1, so the second term is sin(x)*1. So, it's x cos(x) minus sin(x), all over x squared. Yeah, that looks right.Let me see if this matches any of the options given. The options are:A: (x sin x + cos x)/x²B: (x cos x + sin x)/x²C: (x sin x - cos x)/x²D: (x cos x - sin x)/x²So, my result is (x cos x - sin x)/x², which is option D. Just to make sure I didn't make any mistakes, let me think through the steps again. I used the quotient rule correctly, identified u and v properly, took their derivatives correctly, substituted them into the formula, and simplified. Yes, that seems correct.Alternatively, I could also think about using the product rule by rewriting f(x) as sin(x) * (1/x). Then, using the product rule, which is f'(x) = u'(x)v(x) + u(x)v'(x). Let's try that approach to verify.So, if f(x) = sin(x) * (1/x), then u(x) = sin(x) and v(x) = 1/x. Then, u'(x) = cos(x) and v'(x) = -1/x². Applying the product rule: f'(x) = cos(x)*(1/x) + sin(x)*(-1/x²). Simplifying, that's cos(x)/x - sin(x)/x². To combine these terms over a common denominator, which is x², we get [x cos(x) - sin(x)] / x². So, again, I end up with the same result: (x cos x - sin x)/x², which is option D. This confirms that my initial answer was correct.I think I'm confident now that the correct derivative is option D.

💡Okay, so I have this complex number problem here. It says that ( z ) satisfies ( z^2 + 2z = bar{z} ) and ( bar{z} neq z ). I need to find the product of all possible values of ( z ). Hmm, complex numbers can be tricky, but let me try to break it down step by step.First, I remember that any complex number ( z ) can be written as ( a + bi ), where ( a ) and ( b ) are real numbers, and ( i ) is the imaginary unit. The complex conjugate of ( z ), denoted as ( bar{z} ), would then be ( a - bi ). So, if ( z = a + bi ), then ( bar{z} = a - bi ).Given the equation ( z^2 + 2z = bar{z} ), I can substitute ( z ) with ( a + bi ) and ( bar{z} ) with ( a - bi ). Let me write that out:( (a + bi)^2 + 2(a + bi) = a - bi )Okay, now I need to expand ( (a + bi)^2 ). I remember that ( (a + b)^2 = a^2 + 2ab + b^2 ), but since this is a complex number, the ( i ) will come into play. Let me compute it:( (a + bi)^2 = a^2 + 2abi + (bi)^2 )Since ( i^2 = -1 ), this becomes:( a^2 + 2abi - b^2 )So, ( (a + bi)^2 = (a^2 - b^2) + 2abi )Now, let's substitute this back into the original equation:( (a^2 - b^2 + 2abi) + 2(a + bi) = a - bi )Let me simplify the left side by distributing the 2 in the second term:( (a^2 - b^2 + 2abi) + (2a + 2bi) = a - bi )Now, combine like terms:Real parts: ( a^2 - b^2 + 2a )Imaginary parts: ( 2ab + 2b )So, the equation becomes:( (a^2 - b^2 + 2a) + (2ab + 2b)i = a - bi )Since two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal, I can set up a system of equations:1. Real part: ( a^2 - b^2 + 2a = a )2. Imaginary part: ( 2ab + 2b = -b )Let me simplify these equations.Starting with the real part equation:( a^2 - b^2 + 2a = a )Subtract ( a ) from both sides:( a^2 - b^2 + a = 0 )So, equation (1) is:( a^2 + a - b^2 = 0 )Now, the imaginary part equation:( 2ab + 2b = -b )Let me factor out ( b ) on the left side:( b(2a + 2) = -b )Hmm, so ( b(2a + 2) = -b ). Let me bring all terms to one side:( b(2a + 2) + b = 0 )Factor out ( b ):( b(2a + 2 + 1) = 0 )Wait, that doesn't seem right. Let me check my steps again.Original equation:( 2ab + 2b = -b )Bring all terms to the left:( 2ab + 2b + b = 0 )Combine like terms:( 2ab + 3b = 0 )Factor out ( b ):( b(2a + 3) = 0 )Ah, that's better. So, equation (2) is:( b(2a + 3) = 0 )Now, since ( bar{z} neq z ), which means that ( z ) is not a real number, so ( b neq 0 ). Therefore, from equation (2), we have:( 2a + 3 = 0 )Solving for ( a ):( 2a = -3 )( a = -frac{3}{2} )Okay, so ( a = -frac{3}{2} ). Now, let's substitute this back into equation (1) to find ( b ).Equation (1):( a^2 + a - b^2 = 0 )Substitute ( a = -frac{3}{2} ):( left(-frac{3}{2}right)^2 + left(-frac{3}{2}right) - b^2 = 0 )Compute each term:( left(-frac{3}{2}right)^2 = frac{9}{4} )( left(-frac{3}{2}right) = -frac{3}{2} )So, plug these in:( frac{9}{4} - frac{3}{2} - b^2 = 0 )Convert ( frac{3}{2} ) to quarters to combine with ( frac{9}{4} ):( frac{9}{4} - frac{6}{4} - b^2 = 0 )Subtract:( frac{3}{4} - b^2 = 0 )So,( b^2 = frac{3}{4} )Taking square roots:( b = pm frac{sqrt{3}}{2} )Therefore, the possible values of ( z ) are:( z = -frac{3}{2} + frac{sqrt{3}}{2}i ) and ( z = -frac{3}{2} - frac{sqrt{3}}{2}i )Now, the problem asks for the product of all possible values of ( z ). So, I need to compute:( left(-frac{3}{2} + frac{sqrt{3}}{2}iright) times left(-frac{3}{2} - frac{sqrt{3}}{2}iright) )This looks like the product of a complex number and its conjugate, which is equal to the square of the modulus of the complex number. Alternatively, I can compute it directly.Let me compute it directly:Multiply the two complex numbers:( left(-frac{3}{2}right)left(-frac{3}{2}right) + left(-frac{3}{2}right)left(-frac{sqrt{3}}{2}iright) + left(frac{sqrt{3}}{2}iright)left(-frac{3}{2}right) + left(frac{sqrt{3}}{2}iright)left(-frac{sqrt{3}}{2}iright) )Compute each term:1. ( left(-frac{3}{2}right)left(-frac{3}{2}right) = frac{9}{4} )2. ( left(-frac{3}{2}right)left(-frac{sqrt{3}}{2}iright) = frac{3sqrt{3}}{4}i )3. ( left(frac{sqrt{3}}{2}iright)left(-frac{3}{2}right) = -frac{3sqrt{3}}{4}i )4. ( left(frac{sqrt{3}}{2}iright)left(-frac{sqrt{3}}{2}iright) = -frac{3}{4}i^2 )Simplify each term:1. ( frac{9}{4} ) remains as is.2. ( frac{3sqrt{3}}{4}i ) remains as is.3. ( -frac{3sqrt{3}}{4}i ) remains as is.4. ( -frac{3}{4}i^2 ). Since ( i^2 = -1 ), this becomes ( -frac{3}{4}(-1) = frac{3}{4} )Now, add all the terms together:1. ( frac{9}{4} )2. ( frac{3sqrt{3}}{4}i - frac{3sqrt{3}}{4}i = 0 ) (the imaginary parts cancel out)3. ( frac{3}{4} )So, adding the real parts:( frac{9}{4} + frac{3}{4} = frac{12}{4} = 3 )Therefore, the product of all possible values of ( z ) is 3.Wait, let me double-check that. Alternatively, since these are roots of a quadratic equation, the product of the roots is equal to the constant term divided by the leading coefficient. Let me see if I can frame the original equation as a quadratic in ( z ).Starting from ( z^2 + 2z = bar{z} ). Since ( bar{z} ) is the conjugate, and we have ( z ) and ( bar{z} ) in the equation, it's not a standard quadratic equation. However, by expressing ( z ) as ( a + bi ) and solving, we found the two roots. Their product is 3, as calculated.Alternatively, if I consider the quadratic equation in terms of ( z ), but since ( bar{z} ) is involved, it's not straightforward. But since we already found the roots and their product is 3, I think that's correct.So, the product of all possible values of ( z ) is 3.**Final Answer**The product of all possible values of ( z ) is boxed{3}.

💡Alright, so I've got this problem where Jerry wrote a palindrome, and Tom tore it into five parts: MS, SU, US, MUS, and UMM. I need to figure out what the original palindrome was. Okay, let's break this down step by step.First off, a palindrome is a word or phrase that reads the same forwards and backwards. So, whatever Jerry wrote, it should mirror itself. Now, Tom tore it into five parts, and these parts are mixed up. My job is to rearrange these parts to form the original palindrome.Let me list out the parts again to get a clear view:1. MS2. SU3. US4. MUS5. UMMHmm, interesting. I notice that some of these parts are similar or have overlapping letters. For example, MS and SU both have an 'S', and US is just the reverse of SU. MUS is like MS with an extra 'U', and UMM is like UM with an extra 'M'. Maybe these parts can be combined in a way that forms a palindrome.Let me think about how palindromes are structured. They usually have a central point where the word mirrors itself. So, if I can find a central part and build around it, that might work. Let's see if any of these parts can be the center.Looking at UMM, it has two 'M's and a 'U' in the middle. That could potentially be the center of the palindrome since it's symmetrical. If I take UMM as the center, then I need to build around it with the other parts.Now, let's see what we have left: MS, SU, US, and MUS. I notice that MS and SU are both two-letter parts, while MUS is a three-letter part. Maybe MUS can be part of the beginning or the end. Similarly, US is just the reverse of SU, so that might help in mirroring.Wait a second, if I take MUS and reverse it, I get SUM. That's interesting because SU is one of the parts. Maybe I can combine these somehow. Let's try putting MUS at the beginning and SUM at the end. So, MUS...SUM. But I still have MS and SU left.If I place MS at the beginning, it would be MS...SOMETHING...MS reversed. But I already have MUS at the beginning, so maybe that's not the right approach. Let me try another way.What if I consider the entire palindrome? If UMM is the center, then the palindrome would look like something-UMM-something. The something parts should mirror each other. So, if I can find parts that mirror each other around UMM, that would work.Looking at the remaining parts: MS, SU, US, MUS. If I take MUS and reverse it, I get SUM. So, maybe MUS can be on one side and SUM on the other. That would give me MUS-UMM-SUM. Let's see if that works.MUS-UMM-SUM. Putting it all together: MUSUMMSUM. Wait, that's not quite right. Let me check the letters:M U S U M M S U MHmm, that doesn't seem to form a proper palindrome. Maybe I made a mistake in combining them. Let me try again.If I take MUS and place it before UMM, and then place SUM after UMM, it would be MUS-UMM-SUM. But when I write it out, it's MUSUMMSUM. Let me check if this is a palindrome.Reading forwards: M U S U M M S U MReading backwards: M U S M M U S U MWait, that's not the same. The middle part is different. So, MUSUMMSUM is not a palindrome. Hmm, maybe I need to adjust the order.What if I take MUS and place it on one side and US on the other? Let's see: MUS-UMM-US. That would be MUSUMMUS. Let's check:Forwards: M U S U M M U SBackwards: S U M M U S U MNope, that's not the same either. Maybe I need to include all the parts.Wait, I have MS, SU, US, MUS, and UMM. Maybe I need to use all of them. Let's see:If I take MUS and place it on one side, and then place MS and SU on the other side. But I'm not sure how to fit them together.Alternatively, maybe I can interleave them. Let's try:Start with MUS, then add UMM, then add SUM. So, MUS-UMM-SUM. But as I saw before, that doesn't work.Wait, maybe I need to break down the parts differently. Let's see:MUS can be broken down into M, U, S.UMM can be U, M, M.SU is S, U.US is U, S.MS is M, S.Hmm, maybe I can arrange these letters to form a palindrome.Let's list all the letters:From MUS: M, U, SFrom UMM: U, M, MFrom SU: S, UFrom US: U, SFrom MS: M, SSo, total letters:M: 1 (from MUS) + 1 (from UMM) + 1 (from MS) = 3 MsU: 1 (from MUS) + 1 (from UMM) + 1 (from SU) + 1 (from US) = 4 UsS: 1 (from MUS) + 1 (from SU) + 1 (from US) + 1 (from MS) = 4 SsWait, that's 3 Ms, 4 Us, and 4 Ss. But in a palindrome, the number of each letter should be even except possibly one for the center. Here, we have 3 Ms, which is odd, and 4 Us and 4 Ss, which are even. So, the palindrome should have one M in the center and the rest mirrored.So, the structure would be:...M...With mirrored parts on both sides.Given that, let's try to build the palindrome.Starting from the center: MNow, we have 2 Ms left, 4 Us, and 4 Ss.Wait, no, actually, from the total letters:3 Ms, 4 Us, 4 Ss.So, if we have one M in the center, we have 2 Ms left, which can be mirrored on both sides.Similarly, 4 Us and 4 Ss can be mirrored.So, the structure would be:[Part1] M [Part2]Where Part1 and Part2 are mirrors of each other.Now, let's see what parts we have:We have MUS, UMM, SU, US, MS.Wait, but we've already considered the letters. Maybe I need to think differently.Alternatively, maybe the palindrome is longer than just M in the center. Maybe it's longer.Wait, let's think about known palindromes. Sometimes, palindromes are words like "level" or "madam". Maybe this is a longer one.Looking at the parts, MUS and UMM seem like they could be parts of a longer word.Wait, MUS is a word meaning a group of musicians, and UMM is like a sound someone makes. Maybe the palindrome is "SUMMUS".Let me check: S U M M U SYes, that's a palindrome. Let's see if that uses all the parts.Breaking down "SUMMUS":S U M M U SNow, let's see if we can get the parts from this:- SU: S U- MM: M M- US: U S- MUS: M U S- UMM: U M MYes! All the given parts can be found within "SUMMUS".So, the original palindrome Jerry wrote was "SUMMUS".Wait, let me verify:SUMMUSBreaking it down:- SU: positions 1-2: S U- MM: positions 3-4: M M- US: positions 5-6: U S- MUS: positions 2-4: U M M (Wait, no, MUS is M U S, which would be positions 3-5: M U SWait, maybe I need to check differently.Actually, "SUMMUS" can be split as:S U M M U SWhich can be split into:S U, M M, U SBut we also have MUS and UMM as parts.MUS is M U S, which is positions 3-5: M U SUMM is U M M, which is positions 2-4: U M MSo, indeed, all the parts are present within "SUMMUS".Therefore, the original palindrome Jerry wrote was "SUMMUS".

💡Alright, so I have this problem where I need to find three digits to append to the number 456 so that the resulting six-digit number is divisible by 504. Hmm, okay, let's break this down step by step.First, I know that 504 is the number we're dealing with. I remember that 504 can be factored into prime numbers. Let me see, 504 divided by 2 is 252, divided by 2 again is 126, and one more time by 2 is 63. Then, 63 divided by 3 is 21, divided by 3 again is 7, and 7 is a prime number. So, 504 factors into 2^3 * 3^2 * 7. That means for a number to be divisible by 504, it must be divisible by 8, 9, and 7. Okay, that's a useful piece of information.So, the number we're looking at is 456 followed by three unknown digits, let's call them ABC, making it 456ABC. We need 456ABC to be divisible by 504, which means it has to be divisible by 8, 9, and 7.Let me tackle each divisibility rule one by one.Starting with divisibility by 8: A number is divisible by 8 if its last three digits are divisible by 8. In this case, the last three digits are ABC. So, ABC must be divisible by 8.Next, divisibility by 9: A number is divisible by 9 if the sum of its digits is a multiple of 9. So, the sum of all the digits in 456ABC must be divisible by 9. Let's calculate the sum of the known digits first: 4 + 5 + 6 = 15. So, the sum of A, B, and C must be such that 15 + A + B + C is divisible by 9.Lastly, divisibility by 7: This one is a bit trickier. There isn't a straightforward rule like for 8 or 9, but I remember a method where you can take the last digit, double it, subtract it from the rest of the number, and see if the result is divisible by 7. Maybe I can apply that to the entire number 456ABC.But before diving into that, let me see if I can find a way to combine these conditions. Since 504 is 8 * 9 * 7, and these are co-prime factors, the number must satisfy all three conditions simultaneously.So, first, let's handle the divisibility by 8. ABC must be divisible by 8. That gives us a starting point. Let's list all three-digit numbers divisible by 8. The smallest three-digit number divisible by 8 is 104, and the largest is 992. So, ABC can be any number from 104 to 992 that's a multiple of 8.But that's a lot of numbers. Maybe I can narrow it down using the other conditions.Next, the sum of all digits: 15 + A + B + C must be divisible by 9. Let's denote the sum A + B + C as S. So, 15 + S must be divisible by 9. That means S must be congruent to -15 mod 9. Since -15 mod 9 is the same as 3 mod 9 (because -15 + 18 = 3), S must be congruent to 3 mod 9. So, S can be 3, 12, 21, 30, etc. But since A, B, and C are single digits, the maximum S can be is 9 + 9 + 9 = 27. So, possible values for S are 3, 12, 21.So, S can be 3, 12, or 21.Now, let's combine this with the divisibility by 8. We need to find all three-digit numbers ABC divisible by 8, where A + B + C is either 3, 12, or 21.This narrows it down a bit. Let's see how we can approach this.Perhaps I can iterate through all three-digit numbers divisible by 8 and check if their digits sum to 3, 12, or 21. But that might take a while. Maybe there's a smarter way.Alternatively, I can express ABC as 100A + 10B + C. Since ABC is divisible by 8, 100A + 10B + C ≡ 0 mod 8. Also, A + B + C ≡ 3, 12, or 21 mod 9.But since 100 ≡ 4 mod 8, 10 ≡ 2 mod 8, and 1 ≡ 1 mod 8, we can write:100A + 10B + C ≡ 4A + 2B + C ≡ 0 mod 8.So, 4A + 2B + C must be divisible by 8.Additionally, A + B + C must be 3, 12, or 21.This gives us two equations:1. 4A + 2B + C ≡ 0 mod 82. A + B + C = 3, 12, or 21Now, we can try to find all possible combinations of A, B, C (digits from 0 to 9) that satisfy both conditions.Let me consider each possible sum S = 3, 12, 21 and see what combinations of A, B, C can give that sum while also satisfying 4A + 2B + C ≡ 0 mod 8.Starting with S = 3:A + B + C = 3Possible combinations where A, B, C are digits (0-9) and their sum is 3.These are:- A=0, B=0, C=3- A=0, B=1, C=2- A=0, B=2, C=1- A=0, B=3, C=0- A=1, B=0, C=2- A=1, B=1, C=1- A=1, B=2, C=0- A=2, B=0, C=1- A=2, B=1, C=0- A=3, B=0, C=0Now, for each of these combinations, we need to check if 4A + 2B + C is divisible by 8.Let's go through them:1. A=0, B=0, C=3: 4*0 + 2*0 + 3 = 3 → 3 mod 8 ≠ 02. A=0, B=1, C=2: 0 + 2 + 2 = 4 → 4 mod 8 ≠ 03. A=0, B=2, C=1: 0 + 4 + 1 = 5 → 5 mod 8 ≠ 04. A=0, B=3, C=0: 0 + 6 + 0 = 6 → 6 mod 8 ≠ 05. A=1, B=0, C=2: 4 + 0 + 2 = 6 → 6 mod 8 ≠ 06. A=1, B=1, C=1: 4 + 2 + 1 = 7 → 7 mod 8 ≠ 07. A=1, B=2, C=0: 4 + 4 + 0 = 8 → 8 mod 8 = 0 ✅8. A=2, B=0, C=1: 8 + 0 + 1 = 9 → 9 mod 8 = 1 ≠ 09. A=2, B=1, C=0: 8 + 2 + 0 = 10 → 10 mod 8 = 2 ≠ 010. A=3, B=0, C=0: 12 + 0 + 0 = 12 → 12 mod 8 = 4 ≠ 0So, only one combination works for S=3: A=1, B=2, C=0. So, ABC=120.Now, let's move on to S=12.A + B + C = 12This is a bit more involved, but let's think of possible combinations.Since A, B, C are digits from 0 to 9, and their sum is 12, there are many possibilities. Instead of listing all, maybe we can find a way to express this.Alternatively, we can use the equation 4A + 2B + C ≡ 0 mod 8.Let me denote C = 12 - A - B.Substituting into the first equation:4A + 2B + (12 - A - B) ≡ 0 mod 8Simplify:4A + 2B + 12 - A - B ≡ 0 mod 8(4A - A) + (2B - B) + 12 ≡ 0 mod 83A + B + 12 ≡ 0 mod 8So, 3A + B ≡ -12 mod 8But -12 mod 8 is equivalent to 4 mod 8 (since -12 + 16 = 4)So, 3A + B ≡ 4 mod 8Now, we need to find all A and B (digits 0-9) such that 3A + B ≡ 4 mod 8, and A + B ≤ 12 (since C = 12 - A - B must be ≥ 0).Let's consider A from 0 to 9 and find corresponding B.For each A, B must satisfy:B ≡ 4 - 3A mod 8And B must be a digit (0-9), and A + B ≤ 12.Let's go through A from 0 to 9:A=0:B ≡ 4 - 0 ≡ 4 mod 8Possible B: 4, 12 (but 12 is not a digit). So, B=4.Check A + B = 0 + 4 = 4 ≤ 12. Okay.So, C = 12 - 0 - 4 = 8Thus, ABC=048. But leading zero is not allowed in three-digit numbers, so ABC=48, but since we need three digits, it's 048, which is 48. But in the context of appending to 456, it would be 456048. However, leading zeros are generally not considered in such problems unless specified. So, maybe we should consider only A ≥ 1.Wait, actually, in the context of appending digits, A can be zero, making ABC a three-digit number starting with zero, but in reality, that would make it a two-digit number. However, since we're appending three digits, even if the first digit is zero, it's still considered as three digits. So, 048 is acceptable as ABC.But let's see if the problem allows leading zeros. The problem says "three digits," which could imply that leading zeros are allowed. So, we'll include 048 as a possible solution.A=1:B ≡ 4 - 3*1 ≡ 4 - 3 ≡ 1 mod 8Possible B: 1, 9 (since 1 + 8 = 9)Check A + B:For B=1: 1 + 1 = 2 ≤ 12. Okay.C = 12 - 1 - 1 = 10. Wait, C=10 is not a digit. So, invalid.For B=9: 1 + 9 = 10 ≤ 12. Okay.C = 12 - 1 - 9 = 2So, ABC=192A=2:B ≡ 4 - 3*2 ≡ 4 - 6 ≡ -2 ≡ 6 mod 8Possible B: 6, 14 (but 14 is not a digit). So, B=6.Check A + B = 2 + 6 = 8 ≤ 12. Okay.C = 12 - 2 - 6 = 4So, ABC=264A=3:B ≡ 4 - 3*3 ≡ 4 - 9 ≡ -5 ≡ 3 mod 8Possible B: 3, 11 (11 is not a digit). So, B=3.Check A + B = 3 + 3 = 6 ≤ 12. Okay.C = 12 - 3 - 3 = 6So, ABC=336A=4:B ≡ 4 - 3*4 ≡ 4 - 12 ≡ -8 ≡ 0 mod 8Possible B: 0, 8For B=0:A + B = 4 + 0 = 4 ≤ 12. Okay.C = 12 - 4 - 0 = 8So, ABC=408For B=8:A + B = 4 + 8 = 12 ≤ 12. Okay.C = 12 - 4 - 8 = 0So, ABC=480A=5:B ≡ 4 - 3*5 ≡ 4 - 15 ≡ -11 ≡ 5 mod 8Possible B: 5, 13 (13 is not a digit). So, B=5.Check A + B = 5 + 5 = 10 ≤ 12. Okay.C = 12 - 5 - 5 = 2So, ABC=552A=6:B ≡ 4 - 3*6 ≡ 4 - 18 ≡ -14 ≡ 2 mod 8Possible B: 2, 10 (10 is not a digit). So, B=2.Check A + B = 6 + 2 = 8 ≤ 12. Okay.C = 12 - 6 - 2 = 4So, ABC=624A=7:B ≡ 4 - 3*7 ≡ 4 - 21 ≡ -17 ≡ 7 mod 8Possible B: 7, 15 (15 is not a digit). So, B=7.Check A + B = 7 + 7 = 14 > 12. Not allowed.So, no solution for A=7.A=8:B ≡ 4 - 3*8 ≡ 4 - 24 ≡ -20 ≡ 4 mod 8Possible B: 4, 12 (12 is not a digit). So, B=4.Check A + B = 8 + 4 = 12 ≤ 12. Okay.C = 12 - 8 - 4 = 0So, ABC=840A=9:B ≡ 4 - 3*9 ≡ 4 - 27 ≡ -23 ≡ 3 mod 8Possible B: 3, 11 (11 is not a digit). So, B=3.Check A + B = 9 + 3 = 12 ≤ 12. Okay.C = 12 - 9 - 3 = 0So, ABC=930So, for S=12, the possible ABC values are:048, 192, 264, 336, 408, 480, 552, 624, 840, 930Now, let's check each of these to ensure that 4A + 2B + C is divisible by 8.Wait, but we already derived these based on that condition, so they should all satisfy it. But just to be thorough, let's pick a few to verify.Take ABC=192:4*1 + 2*9 + 2 = 4 + 18 + 2 = 24 → 24 mod 8 = 0 ✅ABC=264:4*2 + 2*6 + 4 = 8 + 12 + 4 = 24 → 24 mod 8 = 0 ✅ABC=336:4*3 + 2*3 + 6 = 12 + 6 + 6 = 24 → 24 mod 8 = 0 ✅ABC=408:4*4 + 2*0 + 8 = 16 + 0 + 8 = 24 → 24 mod 8 = 0 ✅ABC=480:4*4 + 2*8 + 0 = 16 + 16 + 0 = 32 → 32 mod 8 = 0 ✅ABC=552:4*5 + 2*5 + 2 = 20 + 10 + 2 = 32 → 32 mod 8 = 0 ✅ABC=624:4*6 + 2*2 + 4 = 24 + 4 + 4 = 32 → 32 mod 8 = 0 ✅ABC=840:4*8 + 2*4 + 0 = 32 + 8 + 0 = 40 → 40 mod 8 = 0 ✅ABC=930:4*9 + 2*3 + 0 = 36 + 6 + 0 = 42 → 42 mod 8 = 2 ≠ 0 ❌Wait, ABC=930: 4*9 + 2*3 + 0 = 36 + 6 + 0 = 42. 42 mod 8 is 42 - 5*8 = 42 - 40 = 2. So, it's not divisible by 8. That's a problem. Did I make a mistake earlier?Yes, I think I did. When A=9, B=3, C=0, we have 4*9 + 2*3 + 0 = 36 + 6 + 0 = 42, which is not divisible by 8. So, ABC=930 doesn't satisfy the divisibility by 8 condition. Therefore, it should be excluded.So, the correct list for S=12 is:048, 192, 264, 336, 408, 480, 552, 624, 840Now, moving on to S=21.A + B + C = 21Again, let's use the same approach.Express C = 21 - A - BSubstitute into 4A + 2B + C ≡ 0 mod 8:4A + 2B + (21 - A - B) ≡ 0 mod 8Simplify:4A + 2B + 21 - A - B ≡ 0 mod 8(4A - A) + (2B - B) + 21 ≡ 0 mod 83A + B + 21 ≡ 0 mod 8So, 3A + B ≡ -21 mod 8-21 mod 8 is equivalent to 5 mod 8 (since -21 + 24 = 3, wait, no: -21 + 24 = 3, but 3 mod 8 is 3. Wait, let me recalculate.Actually, -21 divided by 8 is -2.625, so the remainder is positive. To find -21 mod 8:-21 + 24 = 3, so -21 ≡ 3 mod 8.Wait, no:Wait, 8* (-3) = -24, so -21 - (-24) = 3. So, -21 ≡ 3 mod 8.Therefore, 3A + B ≡ 3 mod 8So, 3A + B ≡ 3 mod 8Again, A and B are digits (0-9), and A + B ≤ 21 (since C = 21 - A - B must be ≥ 0).Let's go through A from 0 to 9 and find corresponding B.For each A, B must satisfy:B ≡ 3 - 3A mod 8And B must be a digit (0-9), and A + B ≤ 21.Let's go through A from 0 to 9:A=0:B ≡ 3 - 0 ≡ 3 mod 8Possible B: 3, 11 (11 is not a digit). So, B=3.Check A + B = 0 + 3 = 3 ≤ 21. Okay.C = 21 - 0 - 3 = 18. Wait, C=18 is not a digit. So, invalid.A=1:B ≡ 3 - 3*1 ≡ 3 - 3 ≡ 0 mod 8Possible B: 0, 8For B=0:A + B = 1 + 0 = 1 ≤ 21. Okay.C = 21 - 1 - 0 = 20. Not a digit. Invalid.For B=8:A + B = 1 + 8 = 9 ≤ 21. Okay.C = 21 - 1 - 8 = 12. Not a digit. Invalid.A=2:B ≡ 3 - 3*2 ≡ 3 - 6 ≡ -3 ≡ 5 mod 8Possible B: 5, 13 (13 is not a digit). So, B=5.Check A + B = 2 + 5 = 7 ≤ 21. Okay.C = 21 - 2 - 5 = 14. Not a digit. Invalid.A=3:B ≡ 3 - 3*3 ≡ 3 - 9 ≡ -6 ≡ 2 mod 8Possible B: 2, 10 (10 is not a digit). So, B=2.Check A + B = 3 + 2 = 5 ≤ 21. Okay.C = 21 - 3 - 2 = 16. Not a digit. Invalid.A=4:B ≡ 3 - 3*4 ≡ 3 - 12 ≡ -9 ≡ 7 mod 8Possible B: 7, 15 (15 is not a digit). So, B=7.Check A + B = 4 + 7 = 11 ≤ 21. Okay.C = 21 - 4 - 7 = 10. Not a digit. Invalid.A=5:B ≡ 3 - 3*5 ≡ 3 - 15 ≡ -12 ≡ 4 mod 8Possible B: 4, 12 (12 is not a digit). So, B=4.Check A + B = 5 + 4 = 9 ≤ 21. Okay.C = 21 - 5 - 4 = 12. Not a digit. Invalid.A=6:B ≡ 3 - 3*6 ≡ 3 - 18 ≡ -15 ≡ 1 mod 8Possible B: 1, 9For B=1:A + B = 6 + 1 = 7 ≤ 21. Okay.C = 21 - 6 - 1 = 14. Not a digit. Invalid.For B=9:A + B = 6 + 9 = 15 ≤ 21. Okay.C = 21 - 6 - 9 = 6So, ABC=696A=7:B ≡ 3 - 3*7 ≡ 3 - 21 ≡ -18 ≡ 6 mod 8Possible B: 6, 14 (14 is not a digit). So, B=6.Check A + B = 7 + 6 = 13 ≤ 21. Okay.C = 21 - 7 - 6 = 8So, ABC=768A=8:B ≡ 3 - 3*8 ≡ 3 - 24 ≡ -21 ≡ 3 mod 8Possible B: 3, 11 (11 is not a digit). So, B=3.Check A + B = 8 + 3 = 11 ≤ 21. Okay.C = 21 - 8 - 3 = 10. Not a digit. Invalid.A=9:B ≡ 3 - 3*9 ≡ 3 - 27 ≡ -24 ≡ 0 mod 8Possible B: 0, 8For B=0:A + B = 9 + 0 = 9 ≤ 21. Okay.C = 21 - 9 - 0 = 12. Not a digit. Invalid.For B=8:A + B = 9 + 8 = 17 ≤ 21. Okay.C = 21 - 9 - 8 = 4So, ABC=984So, for S=21, the possible ABC values are:696, 768, 984Let's verify these:ABC=696:4*6 + 2*9 + 6 = 24 + 18 + 6 = 48 → 48 mod 8 = 0 ✅ABC=768:4*7 + 2*6 + 8 = 28 + 12 + 8 = 48 → 48 mod 8 = 0 ✅ABC=984:4*9 + 2*8 + 4 = 36 + 16 + 4 = 56 → 56 mod 8 = 0 ✅Great, all satisfy the divisibility by 8.Now, compiling all possible ABC values from S=3, S=12, and S=21:From S=3: 120From S=12: 048, 192, 264, 336, 408, 480, 552, 624, 840From S=21: 696, 768, 984But wait, we need to ensure that the entire number 456ABC is divisible by 504, which is 8*9*7. So far, we've ensured divisibility by 8 and 9, but we also need to check divisibility by 7.This adds another layer of complexity. So, even though ABC satisfies the conditions for divisibility by 8 and 9, we still need to check if the entire number 456ABC is divisible by 7.This means we need to check each candidate ABC to see if 456ABC is divisible by 7.This is a bit more involved, but let's proceed.First, let's list all the candidates:From S=3: 120From S=12: 048, 192, 264, 336, 408, 480, 552, 624, 840From S=21: 696, 768, 984So, total candidates: 1 + 9 + 3 = 13Now, let's check each one for divisibility by 7.The number is 456ABC, which is 456000 + ABC.We can write this as 456000 + ABC.To check divisibility by 7, we can use the rule for divisibility by 7, which involves doubling the last digit and subtracting it from the rest of the number, repeating this process until we get a small number that's easy to check.Alternatively, since 456000 is a large number, maybe we can find 456000 mod 7 first, then add ABC mod 7, and see if the total is 0 mod 7.Let's compute 456000 mod 7.First, find 456000 ÷ 7.But instead of dividing directly, let's break it down.We know that 456000 = 456 * 1000Compute 456 mod 7:456 ÷ 7 = 65 * 7 = 455, remainder 1. So, 456 ≡ 1 mod 7Therefore, 456000 ≡ 1 * (1000 mod 7) mod 7Now, compute 1000 mod 7:1000 ÷ 7 = 142 * 7 = 994, remainder 6. So, 1000 ≡ 6 mod 7Thus, 456000 ≡ 1 * 6 ≡ 6 mod 7So, 456000 ≡ 6 mod 7Therefore, 456ABC ≡ 6 + ABC mod 7 ≡ 0 mod 7So, ABC ≡ -6 mod 7 ≡ 1 mod 7 (since -6 + 7 = 1)Therefore, ABC must be congruent to 1 mod 7.So, for each ABC candidate, compute ABC mod 7 and check if it equals 1.Let's go through each candidate:1. ABC=120:120 ÷ 7 = 17 * 7 = 119, remainder 1. So, 120 ≡ 1 mod 7 ✅2. ABC=048:48 ÷ 7 = 6 * 7 = 42, remainder 6. So, 48 ≡ 6 mod 7 ❌3. ABC=192:192 ÷ 7 = 27 * 7 = 189, remainder 3. So, 192 ≡ 3 mod 7 ❌4. ABC=264:264 ÷ 7 = 37 * 7 = 259, remainder 5. So, 264 ≡ 5 mod 7 ❌5. ABC=336:336 ÷ 7 = 48 * 7 = 336, remainder 0. So, 336 ≡ 0 mod 7 ❌6. ABC=408:408 ÷ 7 = 58 * 7 = 406, remainder 2. So, 408 ≡ 2 mod 7 ❌7. ABC=480:480 ÷ 7 = 68 * 7 = 476, remainder 4. So, 480 ≡ 4 mod 7 ❌8. ABC=552:552 ÷ 7 = 78 * 7 = 546, remainder 6. So, 552 ≡ 6 mod 7 ❌9. ABC=624:624 ÷ 7 = 89 * 7 = 623, remainder 1. So, 624 ≡ 1 mod 7 ✅10. ABC=840:840 ÷ 7 = 120 * 7 = 840, remainder 0. So, 840 ≡ 0 mod 7 ❌11. ABC=696:696 ÷ 7 = 99 * 7 = 693, remainder 3. So, 696 ≡ 3 mod 7 ❌12. ABC=768:768 ÷ 7 = 109 * 7 = 763, remainder 5. So, 768 ≡ 5 mod 7 ❌13. ABC=984:984 ÷ 7 = 140 * 7 = 980, remainder 4. So, 984 ≡ 4 mod 7 ❌So, out of all the candidates, only ABC=120 and ABC=624 satisfy ABC ≡ 1 mod 7.Therefore, the possible three-digit numbers to append are 120 and 624.To double-check, let's verify that 456120 and 456624 are indeed divisible by 504.First, 456120:Divide by 504:456120 ÷ 504 = ?Let's compute 504 * 900 = 453,600456,120 - 453,600 = 2,520Now, 504 * 5 = 2,520So, total is 900 + 5 = 905Thus, 504 * 905 = 456,120 ✅Now, 456624:Divide by 504:504 * 900 = 453,600456,624 - 453,600 = 3,024504 * 6 = 3,024So, total is 900 + 6 = 906Thus, 504 * 906 = 456,624 ✅Both numbers are indeed divisible by 504.Therefore, the three digits to append are 120 or 624.

💡Okay, so I have this trigonometry problem here: Given that cos²α equals sinα, I need to find the value of 1/sinα plus cos⁴α. Hmm, let me try to figure this out step by step.First, I remember that there's a fundamental identity in trigonometry that relates sine and cosine: sin²α + cos²α = 1. Maybe I can use that here. Since the problem gives me cos²α = sinα, I can substitute sinα into the identity.So, if cos²α is equal to sinα, then substituting into the identity gives me sin²α + sinα = 1. Wait, is that right? Let me check. The identity is sin²α + cos²α = 1, and since cos²α is sinα, replacing it gives sin²α + sinα = 1. Yeah, that seems correct.Now, I have a quadratic equation in terms of sinα: sin²α + sinα - 1 = 0. Hmm, quadratic equations, okay. Let me write that down: sin²α + sinα - 1 = 0. To solve for sinα, I can use the quadratic formula. The quadratic formula is x = [-b ± sqrt(b² - 4ac)]/(2a). In this case, a is 1, b is 1, and c is -1.Plugging those into the formula: sinα = [-1 ± sqrt(1² - 4*1*(-1))]/(2*1) = [-1 ± sqrt(1 + 4)]/2 = [-1 ± sqrt(5)]/2. So, sinα can be either [-1 + sqrt(5)]/2 or [-1 - sqrt(5)]/2.Wait a second, sine values must be between -1 and 1. Let me compute both solutions to see if they are valid. First, sqrt(5) is approximately 2.236. So, [-1 + 2.236]/2 is about (1.236)/2 ≈ 0.618. That's within the range of sine. The other solution is [-1 - 2.236]/2 ≈ (-3.236)/2 ≈ -1.618, which is less than -1, so that's not possible for a sine value. So, we can discard the negative solution.Therefore, sinα = (-1 + sqrt(5))/2. Got it. Now, the problem asks for 1/sinα + cos⁴α. Let me write that down: 1/sinα + cos⁴α.I already know that cos²α = sinα, so cos⁴α is just (cos²α)², which is (sinα)². So, cos⁴α = sin²α. Therefore, the expression simplifies to 1/sinα + sin²α.So, now I just need to compute 1/sinα + sin²α, where sinα is (-1 + sqrt(5))/2. Let me compute each term separately.First, 1/sinα is 1 divided by (-1 + sqrt(5))/2. That's the same as 2/(-1 + sqrt(5)). Hmm, that denominator has a radical. Maybe I should rationalize it. To rationalize the denominator, I can multiply numerator and denominator by the conjugate of the denominator, which is (-1 - sqrt(5)).So, 2/(-1 + sqrt(5)) * [(-1 - sqrt(5))/(-1 - sqrt(5))] = [2*(-1 - sqrt(5))]/[(-1)^2 - (sqrt(5))^2] = [2*(-1 - sqrt(5))]/[1 - 5] = [2*(-1 - sqrt(5))]/(-4).Simplifying that, the numerator is -2 - 2sqrt(5), and the denominator is -4. So, dividing both numerator terms by -4: (-2)/(-4) = 0.5 and (-2sqrt(5))/(-4) = (sqrt(5))/2. So, altogether, 0.5 + (sqrt(5))/2. Which is (1 + sqrt(5))/2.Wait, let me check that again. So, 2*(-1 - sqrt(5)) is -2 - 2sqrt(5). Divided by -4: (-2 - 2sqrt(5))/(-4) = (2 + 2sqrt(5))/4 = (1 + sqrt(5))/2. Yeah, that's correct.So, 1/sinα is (1 + sqrt(5))/2.Now, let's compute sin²α. Since sinα is (-1 + sqrt(5))/2, squaring that gives [(-1 + sqrt(5))/2]^2. Let me compute that:First, expand the numerator: (-1 + sqrt(5))² = (-1)^2 + 2*(-1)*(sqrt(5)) + (sqrt(5))² = 1 - 2sqrt(5) + 5 = 6 - 2sqrt(5).So, sin²α = (6 - 2sqrt(5))/4. Simplify that by dividing numerator and denominator by 2: (3 - sqrt(5))/2.Okay, so sin²α is (3 - sqrt(5))/2.Now, adding 1/sinα and sin²α together: (1 + sqrt(5))/2 + (3 - sqrt(5))/2.Since the denominators are the same, we can add the numerators: [1 + sqrt(5) + 3 - sqrt(5)]/2 = (4)/2 = 2.So, the value of 1/sinα + cos⁴α is 2.Wait, let me recap to make sure I didn't make any mistakes. Starting from cos²α = sinα, substituted into the Pythagorean identity to get sin²α + sinα - 1 = 0. Solved the quadratic to find sinα = (-1 + sqrt(5))/2. Then, expressed cos⁴α as sin²α. Then, computed 1/sinα by rationalizing the denominator, which gave (1 + sqrt(5))/2. Then, computed sin²α as (3 - sqrt(5))/2. Adding them together: (1 + sqrt(5) + 3 - sqrt(5))/2 = 4/2 = 2. Yeah, that seems correct.I think that makes sense. So, the answer should be 2.**Final Answer**The value of the expression is boxed{2}.

💡Okay, so I have this problem where I need to find the range of values for the expression (frac{1}{x} + frac{1}{y}) given that (x > 0), (y > 0), and (x + 2y = 1). Hmm, let me think about how to approach this.First, I know that since both (x) and (y) are positive, and they satisfy the equation (x + 2y = 1), I can express one variable in terms of the other. Maybe that will help simplify the expression I need to find the range for.Let me try expressing (x) in terms of (y). From the equation (x + 2y = 1), I can subtract (2y) from both sides to get (x = 1 - 2y). But wait, since (x > 0), that means (1 - 2y > 0), so (2y < 1), which implies (y < frac{1}{2}). Also, since (y > 0), the range of (y) is (0 < y < frac{1}{2}).Similarly, if I express (y) in terms of (x), from (x + 2y = 1), I can subtract (x) from both sides to get (2y = 1 - x), so (y = frac{1 - x}{2}). Since (y > 0), this implies (1 - x > 0), so (x < 1). And since (x > 0), the range of (x) is (0 < x < 1).Okay, so now I have expressions for both (x) and (y) in terms of each other. Maybe I can substitute one into the expression (frac{1}{x} + frac{1}{y}) to get it in terms of a single variable.Let me try substituting (x = 1 - 2y) into (frac{1}{x} + frac{1}{y}):[frac{1}{x} + frac{1}{y} = frac{1}{1 - 2y} + frac{1}{y}]Now, this is a function of (y). Let me denote this function as (f(y)):[f(y) = frac{1}{1 - 2y} + frac{1}{y}]I need to find the range of (f(y)) for (0 < y < frac{1}{2}).To find the range, I can analyze the behavior of (f(y)) as (y) approaches the boundaries of its domain and also find any critical points where the function might attain a minimum or maximum.First, let's consider the limits as (y) approaches 0 and (frac{1}{2}):1. As (y to 0^+): - The term (frac{1}{y}) tends to (+infty). - The term (frac{1}{1 - 2y}) approaches 1. - So, (f(y) to +infty).2. As (y to frac{1}{2}^-): - The term (frac{1}{1 - 2y}) tends to (+infty) because the denominator approaches 0. - The term (frac{1}{y}) approaches 2. - So, (f(y) to +infty).So, as (y) approaches either boundary, (f(y)) tends to infinity. That means the function has a minimum somewhere in between. To find this minimum, I can take the derivative of (f(y)) with respect to (y) and set it equal to zero to find critical points.Let's compute the derivative (f'(y)):[f(y) = frac{1}{1 - 2y} + frac{1}{y}][f'(y) = frac{d}{dy}left(frac{1}{1 - 2y}right) + frac{d}{dy}left(frac{1}{y}right)][f'(y) = frac{2}{(1 - 2y)^2} - frac{1}{y^2}]Set (f'(y) = 0) to find critical points:[frac{2}{(1 - 2y)^2} - frac{1}{y^2} = 0][frac{2}{(1 - 2y)^2} = frac{1}{y^2}][2y^2 = (1 - 2y)^2]Let me expand the right-hand side:[2y^2 = 1 - 4y + 4y^2]Bring all terms to one side:[2y^2 - 4y + 1 = 0]This is a quadratic equation in terms of (y). Let me solve for (y) using the quadratic formula:[y = frac{4 pm sqrt{(-4)^2 - 4 cdot 2 cdot 1}}{2 cdot 2}][y = frac{4 pm sqrt{16 - 8}}{4}][y = frac{4 pm sqrt{8}}{4}][y = frac{4 pm 2sqrt{2}}{4}][y = frac{2 pm sqrt{2}}{2}]So, the solutions are:1. (y = frac{2 + sqrt{2}}{2})2. (y = frac{2 - sqrt{2}}{2})But wait, earlier we established that (0 < y < frac{1}{2}). Let's compute the numerical values:1. (y = frac{2 + sqrt{2}}{2} approx frac{2 + 1.4142}{2} = frac{3.4142}{2} approx 1.7071), which is greater than (frac{1}{2}), so it's outside our domain.2. (y = frac{2 - sqrt{2}}{2} approx frac{2 - 1.4142}{2} = frac{0.5858}{2} approx 0.2929), which is less than (frac{1}{2}), so it's within our domain.Therefore, the critical point is at (y = frac{2 - sqrt{2}}{2}).Now, let's compute the corresponding (x) value using (x = 1 - 2y):[x = 1 - 2 cdot frac{2 - sqrt{2}}{2} = 1 - (2 - sqrt{2}) = 1 - 2 + sqrt{2} = -1 + sqrt{2}]But wait, (x) must be positive. Let's check:[sqrt{2} approx 1.4142, so (-1 + 1.4142 approx 0.4142 > 0). Okay, that's fine.So, at (y = frac{2 - sqrt{2}}{2}) and (x = -1 + sqrt{2}), the function (f(y)) attains a critical value. Since the function tends to infinity at both ends of the domain, this critical point must be a minimum.Let's compute the value of (f(y)) at this critical point:[fleft(frac{2 - sqrt{2}}{2}right) = frac{1}{1 - 2 cdot frac{2 - sqrt{2}}{2}} + frac{1}{frac{2 - sqrt{2}}{2}}]Simplify the first term:[1 - 2 cdot frac{2 - sqrt{2}}{2} = 1 - (2 - sqrt{2}) = -1 + sqrt{2}]So, the first term becomes:[frac{1}{-1 + sqrt{2}} = frac{1}{sqrt{2} - 1}]To rationalize the denominator:[frac{1}{sqrt{2} - 1} cdot frac{sqrt{2} + 1}{sqrt{2} + 1} = frac{sqrt{2} + 1}{(sqrt{2})^2 - 1^2} = frac{sqrt{2} + 1}{2 - 1} = sqrt{2} + 1]Now, the second term:[frac{1}{frac{2 - sqrt{2}}{2}} = frac{2}{2 - sqrt{2}}]Again, rationalize the denominator:[frac{2}{2 - sqrt{2}} cdot frac{2 + sqrt{2}}{2 + sqrt{2}} = frac{2(2 + sqrt{2})}{(2)^2 - (sqrt{2})^2} = frac{4 + 2sqrt{2}}{4 - 2} = frac{4 + 2sqrt{2}}{2} = 2 + sqrt{2}]So, adding both terms together:[fleft(frac{2 - sqrt{2}}{2}right) = (sqrt{2} + 1) + (2 + sqrt{2}) = 3 + 2sqrt{2}]Therefore, the minimum value of (frac{1}{x} + frac{1}{y}) is (3 + 2sqrt{2}), and since the function tends to infinity as (y) approaches the boundaries, the range of (frac{1}{x} + frac{1}{y}) is ([3 + 2sqrt{2}, +infty)).I think that's it. Let me just recap to make sure I didn't make any mistakes. I expressed (x) in terms of (y), substituted into the expression, found the derivative, solved for critical points, checked that it's within the domain, computed the value, and confirmed it's a minimum. Seems solid.**Final Answer**The range of values for (frac{1}{x} + frac{1}{y}) is boxed{[3 + 2sqrt{2}, +infty)}.