complete parts a through c for the given function. f(x)=x^(2/3)(x - 4) on -4,4 c. identify the absolute…

complete parts a through c for the given function. f(x)=x^(2/3)(x - 4) on -4,4 c. identify the absolute maximum and minimum values of the function on the given interval (when they exist). select the correct answer boxes to complete your choice. a. the absolute maximum is at x = and there is no absolute minimum. (use a comma to separate answers as needed. type integers or simplified fractions.) b. the absolute minimum is at x = and there is no absolute maximum. (use a comma to separate answers as needed. type integers or simplified fractions.) c. the absolute maximum is at x = and the absolute minimum is at x =. (use a comma to separate answers as needed. type integer or decimals rounded to two decimal places as needed.) d. the function has no absolute extrema.

complete parts a through c for the given function. f(x)=x^(2/3)(x - 4) on -4,4 c. identify the absolute maximum and minimum values of the function on the given interval (when they exist). select the correct answer boxes to complete your choice. a. the absolute maximum is at x = and there is no absolute minimum. (use a comma to separate answers as needed. type integers or simplified fractions.) b. the absolute minimum is at x = and there is no absolute maximum. (use a comma to separate answers as needed. type integers or simplified fractions.) c. the absolute maximum is at x = and the absolute minimum is at x =. (use a comma to separate answers as needed. type integer or decimals rounded to two decimal places as needed.) d. the function has no absolute extrema.

Answer

Explanation:

Step1: Expand the function

$f(x)=x^{\frac{2}{3}}(x - 4)=x^{\frac{5}{3}}-4x^{\frac{2}{3}}$

Step2: Find the derivative

Using the power - rule $(x^n)^\prime=nx^{n - 1}$, we have $f^\prime(x)=\frac{5}{3}x^{\frac{2}{3}}-\frac{8}{3}x^{-\frac{1}{3}}=\frac{5x - 8}{3x^{\frac{1}{3}}}$

Step3: Find the critical points

Set $f^\prime(x) = 0$, then $\frac{5x - 8}{3x^{\frac{1}{3}}}=0$. The numerator gives $5x-8 = 0\Rightarrow x=\frac{8}{5}$, and the denominator gives $x = 0$ (where the derivative is undefined).

Step4: Evaluate the function at critical points and endpoints

  • Evaluate at $x=-4$: $f(-4)=(-4)^{\frac{2}{3}}(-4 - 4)=(-4)^{\frac{2}{3}}\times(-8)$. Since $(-4)^{\frac{2}{3}}=\sqrt[3]{16}$, $f(-4)=-8\sqrt[3]{16}$
  • Evaluate at $x = 0$: $f(0)=0^{\frac{2}{3}}(0 - 4)=0$
  • Evaluate at $x=\frac{8}{5}$: $f(\frac{8}{5})=(\frac{8}{5})^{\frac{2}{3}}(\frac{8}{5}-4)=(\frac{8}{5})^{\frac{2}{3}}\times(-\frac{12}{5})$
  • Evaluate at $x = 4$: $f(4)=4^{\frac{2}{3}}(4 - 4)=0$

We know that $f(-4)\approx-8\times2.52=-20.16$, $f(0) = 0$, $f(\frac{8}{5})=(\frac{8}{5})^{\frac{2}{3}}\times(-\frac{12}{5})\approx1.47\times(-2.4)=-3.53$, $f(4)=0$

Answer:

C. The absolute maximum is $0$ at $x = 0,4$ and the absolute minimum is $-8\sqrt[3]{16}\approx - 20.16$ at $x=-4$