complete parts a through e for the given function. f(x)=x^(2/3)(x - 4) on -4,4 a. locate the critical…

complete parts a through e for the given function. f(x)=x^(2/3)(x - 4) on -4,4 a. locate the critical point(s) of f. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) is(are) at x= (use a comma to separate answers as needed. type an integer or a simplified fraction.) b. the function does not have a critical value.

complete parts a through e for the given function. f(x)=x^(2/3)(x - 4) on -4,4 a. locate the critical point(s) of f. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) is(are) at x= (use a comma to separate answers as needed. type an integer or a simplified fraction.) b. the function does not have a critical value.

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 $\frac{d}{dx}(x^n)=nx^{n - 1}$, 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: Set the derivative equal to zero

$\frac{5x - 8}{3x^{\frac{1}{3}}}=0$. The numerator must be zero for the fraction to be zero. So $5x-8 = 0$, which gives $x=\frac{8}{5}$

Step4: Consider where the derivative is undefined

The derivative $f^\prime(x)$ is undefined when $x = 0$ since the denominator $3x^{\frac{1}{3}}=0$ at $x = 0$

Answer:

A. The critical point(s) is(are) at $x = 0,\frac{8}{5}$