find the critical numbers of the function. (enter your answers as a comma - separated list. if an answer…

Explanation:

Step1: Use product - rule to find derivative

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = x^{4/5}$ and $v=(x - 4)^2$. First, find $u^\prime$ and $v^\prime$. $u^\prime=\frac{4}{5}x^{4/5 - 1}=\frac{4}{5}x^{-1/5}$, and $v^\prime = 2(x - 4)$. Then $F^\prime(x)=\frac{4}{5}x^{-1/5}(x - 4)^2+x^{4/5}\cdot2(x - 4)$.

Step2: Factor out common factors

$F^\prime(x)=(x - 4)\left(\frac{4}{5}x^{-1/5}(x - 4)+2x^{4/5}\right)$. $=(x - 4)\left(\frac{4(x - 4)}{5x^{1/5}}+2x^{4/5}\right)$. Get a common denominator: $=(x - 4)\left(\frac{4x-16 + 10x}{5x^{1/5}}\right)=(x - 4)\left(\frac{14x-16}{5x^{1/5}}\right)$.

Step3: Set the derivative equal to zero

Set $F^\prime(x)=0$. Then either $x - 4 = 0$ or $14x-16 = 0$ or $x = 0$ (since the derivative is undefined at $x = 0$). If $x - 4=0$, then $x = 4$. If $14x-16 = 0$, then $14x=16$, and $x=\frac{8}{7}$. The derivative is undefined at $x = 0$.

Answer:

$0,\frac{8}{7},4$