💡First, I recognize that (64) is a perfect cube, specifically (4^3). This allows me to set the cube of (2x - 8) equal to (4^3).Next, I take the cube root of both sides of the equation to eliminate the cube on the left side, simplifying the equation to (2x - 8 = 4).Then, I solve for (x) by isolating the variable. I add (8) to both sides to get (2x = 12), and finally, I divide both sides by (2) to find that (x = 6).

💡Okay, so I have this problem about an odd function f(x) defined on all real numbers. The problem says that for any 0 < x₁ < x₂, the difference quotient (f(x₂) - f(x₁))/(x₂ - x₁) is less than 3. Also, it's given that f(3) = 9. I need to find the solution set for the inequality f(x) - 3x < 0. The options are A, B, C, D, which are different intervals.First, let me recall what an odd function is. An odd function satisfies f(-x) = -f(x) for all x. So, the graph of an odd function is symmetric about the origin. That might be useful later.Now, the condition given is about the difference quotient being less than 3 for any two points x₁ and x₂ where 0 < x₁ < x₂. That sounds like it's related to the derivative, but it's not exactly the derivative because it's just the slope between two points, not the limit as the points get close. But maybe I can think of it as the function's average rate of change being less than 3 on any interval in the positive x-axis.So, if I rearrange the inequality (f(x₂) - f(x₁))/(x₂ - x₁) < 3, it becomes f(x₂) - f(x₁) < 3(x₂ - x₁). That simplifies to f(x₂) - 3x₂ < f(x₁) - 3x₁. Hmm, that looks like the function f(x) - 3x is decreasing on the interval (0, ∞). Because if x₂ > x₁, then f(x₂) - 3x₂ < f(x₁) - 3x₁, which means the function is decreasing.Let me define a new function g(x) = f(x) - 3x. Then, from the inequality above, g(x₂) < g(x₁) whenever x₂ > x₁ in (0, ∞). So, g(x) is decreasing on (0, ∞).Since f(x) is odd, let's see what happens with g(-x). g(-x) = f(-x) - 3(-x) = -f(x) + 3x = -(f(x) - 3x) = -g(x). So, g(x) is also an odd function. That means the behavior of g(x) on the negative side is just the mirror image of its behavior on the positive side.Given that g(x) is decreasing on (0, ∞), and it's odd, what does that say about its behavior on (-∞, 0)? Well, if g(x) is decreasing on the positive side, then on the negative side, since it's odd, it should also be decreasing. Wait, let me think about that. If g(x) is decreasing on the positive side, then as x increases, g(x) decreases. On the negative side, as x increases (towards zero), g(x) would also decrease because it's odd. So, yes, g(x) is decreasing on both sides.Now, we know that f(3) = 9. Let's compute g(3). g(3) = f(3) - 3*3 = 9 - 9 = 0. So, g(3) = 0. Since g(x) is odd, g(-3) = -g(3) = 0. So, g(-3) is also 0.Now, the inequality we need to solve is f(x) - 3x < 0, which is equivalent to g(x) < 0.So, we need to find all x where g(x) < 0.Since g(x) is decreasing on (0, ∞) and g(3) = 0, that means for x > 3, g(x) will be less than g(3) = 0 because it's decreasing. Similarly, for x between 0 and 3, since g(3) = 0 and it's decreasing, g(x) will be greater than 0 in that interval because as x increases, g(x) decreases from some higher value to 0 at x=3.Wait, hold on. If g(x) is decreasing on (0, ∞), then as x increases, g(x) decreases. So, at x=0, what is g(0)? Since f(x) is odd, f(0) = 0. So, g(0) = f(0) - 3*0 = 0. So, g(0) = 0.But wait, if g(x) is decreasing on (0, ∞), starting at g(0)=0, then as x increases, g(x) decreases. So, at x=3, g(3)=0. That means between x=0 and x=3, g(x) is decreasing from 0 to 0? That doesn't make sense because if it's decreasing, it should go from higher to lower. Hmm, maybe I made a mistake.Wait, f(3)=9, so g(3)=0. Since g(x) is decreasing on (0, ∞), starting from g(0)=0, it must be that g(x) is negative for x > 3 and positive for x between 0 and 3. Because if it's decreasing, starting at 0, it goes down as x increases. So, between 0 and 3, it's still above 0, but after 3, it's below 0.Similarly, on the negative side, since g(x) is odd, it's also decreasing. So, at x=-3, g(-3)=0. For x between -3 and 0, since it's decreasing, g(x) would be increasing from g(-3)=0 to g(0)=0? Wait, that doesn't make sense either.Wait, no. If g(x) is decreasing on (-∞, 0), then as x increases towards 0, g(x) decreases. So, at x=-3, g(-3)=0. As x increases from -∞ to 0, g(x) decreases. So, for x < -3, g(x) would be greater than 0, and for x between -3 and 0, g(x) would be less than 0.Wait, let me think again. If g(x) is decreasing on (-∞, 0), then as x increases, g(x) decreases. So, starting from x=-∞, g(x) is approaching some limit, and as x increases towards 0, g(x) decreases. At x=-3, g(-3)=0. So, for x < -3, since g(x) is decreasing as x increases, g(x) must be greater than 0. For x between -3 and 0, g(x) is less than 0.So, putting it all together:- For x > 3, g(x) < 0- For 0 < x < 3, g(x) > 0- For x = 3, g(x)=0- For x between -3 and 0, g(x) < 0- For x < -3, g(x) > 0- At x=0, g(x)=0Therefore, the inequality g(x) < 0 holds for x in (-3, 0) and x > 3. So, the solution set is (-3, 0) union (3, ∞).Looking back at the options:A: (-∞, -3) ∪ (3, +∞)B: (-3, 0) ∪ (3, +∞)C: (-∞, -3) ∪ (0, 3)D: (-3, 0) ∪ (0, 3)So, the correct answer should be B: (-3, 0) ∪ (3, +∞).Wait, but let me double-check. I concluded that g(x) < 0 for x in (-3, 0) and x > 3. So, yes, that's option B.I think that's correct.

💡First, I'll assign variables to represent the ages of A, B, and C.Let A be the age of person A, B be the age of person B, and C be the age of person C.From the problem, I know that the total age of A and B is greater than the total age of B and C. This can be expressed as:A + B > B + CAdditionally, it's given that C is 12 years younger than A:C = A - 12To find out how many years more the total age of A and B is compared to the total age of B and C, I'll calculate the difference:Difference = (A + B) - (B + C)Substituting the second equation into the first:Difference = (A + B) - (B + (A - 12))Simplifying the equation:Difference = A + B - B - A + 12Difference = 12Therefore, the total age of A and B is 12 years more than the total age of B and C.

💡Alright, so I have this problem about a parabola and a point, and I need to find the slope of a line that passes through the focus of the parabola. The condition given is that the vectors from the point M to the intersection points A and B are perpendicular, which means their dot product is zero. Hmm, okay, let me try to break this down step by step.First, the parabola is given by the equation ( y^2 = 8x ). I remember that the standard form of a parabola that opens to the right is ( y^2 = 4px ), where p is the distance from the vertex to the focus. Comparing this to the given equation, ( 4p = 8 ), so ( p = 2 ). That means the focus of the parabola is at (2, 0). Got that down.Next, the point M is given as (-2, 2). So, we have this point M, and a line passing through the focus (2, 0) with a slope k intersects the parabola at points A and B. The condition is that the vectors MA and MB are perpendicular, which translates to their dot product being zero. That's the key condition here.Okay, so let's think about how to approach this. I need to find the equation of the line passing through the focus with slope k. The general equation of a line with slope k passing through a point (x0, y0) is ( y - y0 = k(x - x0) ). Substituting the focus (2, 0), the equation becomes ( y = k(x - 2) ). So, that's our line.Now, this line intersects the parabola ( y^2 = 8x ). To find the points of intersection, I can substitute the expression for y from the line into the parabola equation. Let's do that:( [k(x - 2)]^2 = 8x )Expanding that:( k^2(x - 2)^2 = 8x )( k^2(x^2 - 4x + 4) = 8x )( k^2x^2 - 4k^2x + 4k^2 = 8x )Bring all terms to one side:( k^2x^2 - (4k^2 + 8)x + 4k^2 = 0 )So, that's a quadratic equation in terms of x. Let's denote this as:( ax^2 + bx + c = 0 )Where:- ( a = k^2 )- ( b = -(4k^2 + 8) )- ( c = 4k^2 )Now, since the line intersects the parabola at points A and B, let's denote their coordinates as ( A(x_1, y_1) ) and ( B(x_2, y_2) ). From the quadratic equation, we can use Vieta's formulas to find the sum and product of the roots.Sum of roots:( x_1 + x_2 = frac{-b}{a} = frac{4k^2 + 8}{k^2} = 4 + frac{8}{k^2} )Product of roots:( x_1x_2 = frac{c}{a} = frac{4k^2}{k^2} = 4 )Okay, so we have ( x_1 + x_2 = 4 + frac{8}{k^2} ) and ( x_1x_2 = 4 ).Now, since points A and B lie on the line ( y = k(x - 2) ), their y-coordinates can be expressed in terms of their x-coordinates:( y_1 = k(x_1 - 2) )( y_2 = k(x_2 - 2) )So, let's find ( y_1 + y_2 ) and ( y_1y_2 ):Sum of y-coordinates:( y_1 + y_2 = k(x_1 - 2) + k(x_2 - 2) = k(x_1 + x_2 - 4) )We already know ( x_1 + x_2 = 4 + frac{8}{k^2} ), so:( y_1 + y_2 = kleft(4 + frac{8}{k^2} - 4right) = kleft(frac{8}{k^2}right) = frac{8}{k} )Product of y-coordinates:( y_1y_2 = [k(x_1 - 2)][k(x_2 - 2)] = k^2(x_1 - 2)(x_2 - 2) )Let's expand ( (x_1 - 2)(x_2 - 2) ):( x_1x_2 - 2x_1 - 2x_2 + 4 )We know ( x_1x_2 = 4 ) and ( x_1 + x_2 = 4 + frac{8}{k^2} ), so:( (x_1 - 2)(x_2 - 2) = 4 - 2(x_1 + x_2) + 4 = 8 - 2left(4 + frac{8}{k^2}right) = 8 - 8 - frac{16}{k^2} = -frac{16}{k^2} )Therefore, the product of y-coordinates:( y_1y_2 = k^2 times left(-frac{16}{k^2}right) = -16 )Alright, so now we have:- ( x_1 + x_2 = 4 + frac{8}{k^2} )- ( x_1x_2 = 4 )- ( y_1 + y_2 = frac{8}{k} )- ( y_1y_2 = -16 )Now, the condition given is that vectors MA and MB are perpendicular, which means their dot product is zero. Let's express vectors MA and MB.Point M is (-2, 2). So, vector MA is from M to A: ( (x_1 - (-2), y_1 - 2) = (x_1 + 2, y_1 - 2) )Similarly, vector MB is from M to B: ( (x_2 + 2, y_2 - 2) )The dot product of MA and MB is:( (x_1 + 2)(x_2 + 2) + (y_1 - 2)(y_2 - 2) = 0 )Let's compute each part separately.First, expand ( (x_1 + 2)(x_2 + 2) ):( x_1x_2 + 2x_1 + 2x_2 + 4 )We know ( x_1x_2 = 4 ) and ( x_1 + x_2 = 4 + frac{8}{k^2} ), so:( 4 + 2left(4 + frac{8}{k^2}right) + 4 = 4 + 8 + frac{16}{k^2} + 4 = 16 + frac{16}{k^2} )Wait, hold on, that doesn't seem right. Let me recalculate:Wait, ( (x_1 + 2)(x_2 + 2) = x_1x_2 + 2x_1 + 2x_2 + 4 )We know ( x_1x_2 = 4 ), ( x_1 + x_2 = 4 + frac{8}{k^2} ), so:( 4 + 2(x_1 + x_2) + 4 = 4 + 2left(4 + frac{8}{k^2}right) + 4 )Compute that:4 + 8 + ( frac{16}{k^2} ) + 4 = 16 + ( frac{16}{k^2} )Wait, that seems correct. So, the first part is ( 16 + frac{16}{k^2} )Now, the second part is ( (y_1 - 2)(y_2 - 2) ). Let's expand that:( y_1y_2 - 2y_1 - 2y_2 + 4 )We know ( y_1y_2 = -16 ), ( y_1 + y_2 = frac{8}{k} ), so:( -16 - 2(y_1 + y_2) + 4 = -16 - 2left(frac{8}{k}right) + 4 = -12 - frac{16}{k} )So, putting it all together, the dot product is:( left(16 + frac{16}{k^2}right) + left(-12 - frac{16}{k}right) = 0 )Simplify:( 16 + frac{16}{k^2} - 12 - frac{16}{k} = 0 )Combine like terms:( (16 - 12) + frac{16}{k^2} - frac{16}{k} = 0 )Which simplifies to:( 4 + frac{16}{k^2} - frac{16}{k} = 0 )Let me write that as:( frac{16}{k^2} - frac{16}{k} + 4 = 0 )Hmm, this is a quadratic equation in terms of ( frac{1}{k} ). Let me set ( u = frac{1}{k} ), so the equation becomes:( 16u^2 - 16u + 4 = 0 )Divide all terms by 4 to simplify:( 4u^2 - 4u + 1 = 0 )Now, let's solve for u:Using the quadratic formula, ( u = frac{4 pm sqrt{(-4)^2 - 4 times 4 times 1}}{2 times 4} )Compute discriminant:( 16 - 16 = 0 )So, discriminant is zero, meaning one real solution:( u = frac{4}{8} = frac{1}{2} )So, ( u = frac{1}{2} ), which means ( frac{1}{k} = frac{1}{2} ), so ( k = 2 ).Wait, so k is 2. Let me check if that makes sense.Let me verify the calculations to make sure I didn't make any mistakes.Starting from the dot product:( (x_1 + 2)(x_2 + 2) + (y_1 - 2)(y_2 - 2) = 0 )Computed ( (x_1 + 2)(x_2 + 2) = 16 + frac{16}{k^2} )Computed ( (y_1 - 2)(y_2 - 2) = -12 - frac{16}{k} )Adding them together:( 16 + frac{16}{k^2} - 12 - frac{16}{k} = 4 + frac{16}{k^2} - frac{16}{k} = 0 )Then, setting ( u = frac{1}{k} ), equation becomes:( 16u^2 - 16u + 4 = 0 )Divide by 4: ( 4u^2 - 4u + 1 = 0 )Quadratic in u: discriminant is 16 - 16 = 0, so u = 4/(2*4) = 0.5, so u = 1/2, so k = 2.Yes, that seems consistent.Wait, but let me think about whether k=2 is one of the options. The options are:A: 1/2B: sqrt(2)/2C: sqrt(2)D: 2Yes, D is 2, so that's the answer.But just to be thorough, let me plug k=2 back into the original equations and see if everything holds.First, the line is ( y = 2(x - 2) ), which is ( y = 2x - 4 ).Substitute into parabola ( y^2 = 8x ):( (2x - 4)^2 = 8x )Expand:( 4x^2 - 16x + 16 = 8x )Bring all terms to one side:( 4x^2 - 24x + 16 = 0 )Divide by 4:( x^2 - 6x + 4 = 0 )Solutions:( x = [6 ± sqrt(36 - 16)] / 2 = [6 ± sqrt(20)] / 2 = [6 ± 2sqrt(5)] / 2 = 3 ± sqrt(5) )So, x1 = 3 + sqrt(5), x2 = 3 - sqrt(5)Then, y1 = 2(x1 - 2) = 2(1 + sqrt(5)) = 2 + 2sqrt(5)y2 = 2(x2 - 2) = 2(1 - sqrt(5)) = 2 - 2sqrt(5)Now, points A and B are (3 + sqrt(5), 2 + 2sqrt(5)) and (3 - sqrt(5), 2 - 2sqrt(5))Point M is (-2, 2)Compute vectors MA and MB:Vector MA: (3 + sqrt(5) - (-2), 2 + 2sqrt(5) - 2) = (5 + sqrt(5), 2sqrt(5))Vector MB: (3 - sqrt(5) - (-2), 2 - 2sqrt(5) - 2) = (5 - sqrt(5), -2sqrt(5))Compute the dot product:(5 + sqrt(5))(5 - sqrt(5)) + (2sqrt(5))(-2sqrt(5))First term: 25 - (sqrt(5))^2 = 25 - 5 = 20Second term: -4*5 = -20Total: 20 - 20 = 0Yes, the dot product is zero, so the vectors are perpendicular. So, k=2 is indeed the correct answer.I think that's thorough enough. I didn't make any calculation errors, and plugging back in confirms the result.**Final Answer**boxed{D}

💡First, I need to determine the initial number of regular members in the club. Since there are 3 board members and a total of 9 people, the number of regular members is 9 - 3 = 6.Each year, every regular member recruits one new person, effectively doubling the number of regular members each year. The board members retire and are not replaced, so they do not contribute to the growth.I will calculate the number of regular members for each year:- Year 1: 6 regular members * 2 = 12 regular members- Year 2: 12 regular members * 2 = 24 regular members- Year 3: 24 regular members * 2 = 48 regular members- Year 4: 48 regular members * 2 = 96 regular membersAfter four years, there are 96 regular members in the club. Since the board members are not replaced, the total number of people in the club is equal to the number of regular members.

💡Okay, so I have this problem here: For positive numbers x and y, if 1/x + 4/y = 1 and the inequality x + y < m² - 8m has a solution, then I need to find the range of real number m. Hmm, let me try to figure this out step by step.First, I know that x and y are positive numbers. The equation given is 1/x + 4/y = 1. I need to relate this to the inequality x + y < m² - 8m. Maybe I can find the minimum value of x + y given the equation, and then set that minimum to be less than m² - 8m. If that's possible, then I can find the range of m.Let me write down the equation again: 1/x + 4/y = 1. I want to express x + y in terms that I can minimize. Maybe I can use the method of Lagrange multipliers or some inequality like Cauchy-Schwarz or AM-GM. Since the problem involves reciprocals, perhaps AM-GM is the way to go.Wait, another approach is to express y in terms of x or vice versa from the equation and substitute into x + y. Let me try that.From 1/x + 4/y = 1, let's solve for y:1/x + 4/y = 1Subtract 1/x from both sides:4/y = 1 - 1/xWhich is 4/y = (x - 1)/xThen, take reciprocal:y/4 = x/(x - 1)So, y = 4x/(x - 1)Hmm, okay. So y is expressed in terms of x. Now, since x and y are positive, the denominator x - 1 must be positive because y is positive. So x - 1 > 0 => x > 1.So, x must be greater than 1.Now, let's write x + y:x + y = x + 4x/(x - 1)Let me combine these terms:x + 4x/(x - 1) = [x(x - 1) + 4x]/(x - 1) = [x² - x + 4x]/(x - 1) = [x² + 3x]/(x - 1)So, x + y = (x² + 3x)/(x - 1)Hmm, now I need to find the minimum value of this expression for x > 1.Alternatively, maybe I can use calculus here. Let me set f(x) = (x² + 3x)/(x - 1). Then, find its derivative and set it to zero to find critical points.First, compute f'(x):Using the quotient rule: f'(x) = [ (2x + 3)(x - 1) - (x² + 3x)(1) ] / (x - 1)²Let me compute the numerator:(2x + 3)(x - 1) = 2x(x) - 2x(1) + 3(x) - 3(1) = 2x² - 2x + 3x - 3 = 2x² + x - 3Subtract (x² + 3x):2x² + x - 3 - x² - 3x = x² - 2x - 3So, f'(x) = (x² - 2x - 3)/(x - 1)²Set f'(x) = 0:x² - 2x - 3 = 0Solving this quadratic equation:x = [2 ± sqrt(4 + 12)] / 2 = [2 ± sqrt(16)] / 2 = [2 ± 4]/2So, x = (2 + 4)/2 = 6/2 = 3, or x = (2 - 4)/2 = (-2)/2 = -1But x must be greater than 1, so x = 3 is the critical point.Now, let's check the second derivative or use test points to see if this is a minimum.Alternatively, since we're dealing with positive x > 1, and the function f(x) tends to infinity as x approaches 1 from the right and as x approaches infinity, the critical point at x = 3 is likely a minimum.So, let's compute f(3):f(3) = (9 + 9)/(3 - 1) = 18/2 = 9So, the minimum value of x + y is 9.Therefore, x + y >= 9, with equality when x = 3 and y = 6 (since y = 4x/(x - 1) = 4*3/(2) = 6).Now, the inequality given is x + y < m² - 8m. For this inequality to have a solution, there must exist some x and y such that x + y is less than m² - 8m. But since the minimum of x + y is 9, we need 9 < m² - 8m.So, m² - 8m > 9Which is m² - 8m - 9 > 0Let me solve this quadratic inequality.First, find the roots of m² - 8m - 9 = 0.Using quadratic formula:m = [8 ± sqrt(64 + 36)] / 2 = [8 ± sqrt(100)] / 2 = [8 ± 10]/2So, m = (8 + 10)/2 = 18/2 = 9, or m = (8 - 10)/2 = (-2)/2 = -1So, the quadratic can be factored as (m - 9)(m + 1) > 0Now, to solve this inequality, we consider the intervals determined by the roots: (-∞, -1), (-1, 9), and (9, ∞).Test each interval:1. For m < -1, say m = -2: (-2 - 9)(-2 + 1) = (-11)(-1) = 11 > 02. For -1 < m < 9, say m = 0: (0 - 9)(0 + 1) = (-9)(1) = -9 < 03. For m > 9, say m = 10: (10 - 9)(10 + 1) = (1)(11) = 11 > 0So, the inequality (m - 9)(m + 1) > 0 holds when m < -1 or m > 9.Therefore, the range of real number m is (-∞, -1) ∪ (9, ∞).Wait, let me just double-check my steps to make sure I didn't make any mistakes.1. Starting from 1/x + 4/y = 1, I solved for y and got y = 4x/(x - 1). That seems correct.2. Then, I expressed x + y as (x² + 3x)/(x - 1). Let me verify that:x + y = x + 4x/(x - 1) = [x(x - 1) + 4x]/(x - 1) = [x² - x + 4x]/(x - 1) = [x² + 3x]/(x - 1). Yes, that's correct.3. Then, I took the derivative f'(x) using the quotient rule. Let me recheck that:f'(x) = [ (2x + 3)(x - 1) - (x² + 3x)(1) ] / (x - 1)²Expanding numerator:(2x + 3)(x - 1) = 2x² - 2x + 3x - 3 = 2x² + x - 3Subtract (x² + 3x): 2x² + x - 3 - x² - 3x = x² - 2x - 3. Correct.4. Solving x² - 2x - 3 = 0 gives x = 3 and x = -1. Since x > 1, x = 3 is the critical point. Correct.5. Calculating f(3) = (9 + 9)/(3 - 1) = 18/2 = 9. Correct.6. So, x + y >= 9, so 9 < m² - 8m. Correct.7. Solving m² - 8m - 9 > 0 gives roots at m = 9 and m = -1. Testing intervals shows that the inequality holds for m < -1 and m > 9. Correct.Therefore, the range of m is (-∞, -1) ∪ (9, ∞).**Final Answer**The range of real number ( m ) is boxed{(-infty, -1) cup (9, +infty)}.