Question

find the critical values of the function (f(x)=x^{1/5}-x^{-4/5}). answer (separate by commas, or enter
one\ (without quotes) if a critical value doesnt exist): (x =)

Explanation:

Step1: Find the derivative of the function

Use the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$. For $y = f(x)=x^{\frac{1}{5}}-x^{-\frac{4}{5}}$, the derivative $f'(x)=\frac{1}{5}x^{-\frac{4}{5}}+\frac{4}{5}x^{-\frac{9}{5}}=\frac{1}{5x^{\frac{4}{5}}}+\frac{4}{5x^{\frac{9}{5}}}=\frac{x + 4}{5x^{\frac{9}{5}}}$.

Step2: Set the derivative equal to zero and find where it is undefined

Set $f'(x)=0$, then $\frac{x + 4}{5x^{\frac{9}{5}}}=0$. The numerator $x+4 = 0$ gives $x=-4$. The derivative $f'(x)$ is undefined when $x = 0$ since the denominator $5x^{\frac{9}{5}}=0$ at $x = 0$.

Answer:

$-4,0$