Chapter 3, Problem 55RE

To determine

To find: The given function is differentiable, continuous but not differentiable, and neither continuous nor differentiable.

Answer to Problem 55RE

Differentiable at $x=\left(-\infty ,0\right)\cup \left(0,\infty \right)$

Continuous but not differentiable at $x=0$.

Neither continuous nor differentiable at none.

Explanation of Solution

Given information: $y=x^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$.

The given $y=x^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$ function is continuous over its entire interval.

  $\begin{array}{l}\frac{dy}{dx}=\frac{d}{dx}\left(x^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)\\ \frac{dy}{dx}=\frac{d}{dx}\left(\frac{x^4}{x^5}\right)\\ \frac{dy}{dx}=\frac{4}{5x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}\end{array}$

The function is not differentiable at $x=0$.

Conclusion:

The given function is differentiable at $x=\left(-\infty ,0\right)u\left(0,\infty \right)$

The function is continuous but not differentiable at $x=0$.

The function is neither continuous nor differentiable at nowhere.