What is the absolute maximum value of f ( x ) = x 2 x − 7 3 {"version":"1.1","math":"f(x)=x^2\sqrt[3]{x-7}"}
on [ − 4 , 8 ] ? {"version":"1.1","math":"[-4,8]?"}

Question

What is the absolute maximum value of  f ( x ) =  x 2    x − 7  3  {"version":"1.1","math":"f(x)=x^2\sqrt[3]{x-7}"}
on  [ − 4 , 8 ] ? {"version":"1.1","math":"[-4,8]?"}

BlackTom AI · Accepted Answer

We are given the function f(x) = x^2 * cube_root(x - 7) on the closed interval [-4, 8], and asked for the absolute maximum value.

First, identify the critical points inside the interval by differentiating f. Let f(x) = x^2 (x - 7)^{1/3}. Using the product rule: f'(x) = 2x (x - 7)^{1/3} + x^2 * (1/3) (x - 7)^{-2/3}.

To find where f'(x) = 0, clear the fractional exponents by multiplying by 3 (x - 7)^{2/3} (which is valid for x ≠ 7; if x = 7, we can check separately). This gives:
  3 * f'(x) * (x - 7)^{2/3} = 6x (x - 7) + x^2 = 0.
So we solve 6x(x - 7) + x^2 = 0, which simplifies to 7x^2 - 42x = 0, i.e. 7x(x - 6) = 0.
Thus the critical points inside the domain are x = 0 and x = 6. We also consider the endpoints x = -4 and x = 8.

Now evaluate f at these points:
- At x = -4: f(-4) = (-4)^2 * cube_root(-4 - 7) = 16 * cube_root(-11) ≈ 16 * (-2.22398) ≈ -35.58.
- At x = 0: f(0) = 0^2 * cube_root(0 - 7) = 0 * cube_root(-7) = 0.
- At x = 6: f(6) = 6^2 * cube_root(6 - 7) = 36 * cube_root(-1) = 36 * (-1) = -36.
- At x = 8: f(8) = 8^2 * cube_root(8 - 7) = 64 * cube_root(1) = 64 * 1 = 64.

Comparing these values, the largest among the computed f(x) values on the interval is the one at x = 8 with f(8) = 64. The negative values at the interior points and at the left endpoint are smaller, and the value at x = 0 is 0, which is still less than 64.

Answer-check notes: if the provided multiple-choice option appears as a non-numeric label such as "Answer" rather than a numeric value, it does not convey the numeric maximum and cannot be the correct choice for the maximum value. In such a case, the actual maximum value arises from evaluating the critical points and endpoints as shown above.

Summary of reasoning steps:
- Differentiate f(x) and solve f'(x) = 0 to find critical points: x = 0 and x = 6.
- Evaluate f at all candidates: x = -4, 0, 6, 8.
- Compare resulting values: -35.58..., 0, -36, 64; the largest is 64 at x = 8.

What is the absolute maximum value of f ( x ) = x 2 x − 7 3 {"version":"1.1","math":"f(x)=x^2\sqrt[3]{x-7}"} on [ − 4 , 8 ] ? {"version":"1.1","math":"[-4,8]?"}

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