Chapter 1.5, Problem 48E

To determine

Find the graph of the given function ad visually determine the open intervals on which the given function is increasing, decreasing or constant.

Answer to Problem 48E

The function $f\left(x\right)$ is decreasing on the interval $\left(–\infty ,0\right)$ and increasing on the open interval $\left(0,\infty \right).$

Explanation of Solution

Given:

The equation is $f(x)=x^{\frac{2}{3}}$

The equation is $f(x)=x^{\frac{2}{3}}$

Use the graphing utility for $f\left(x\right)$

The function $f\left(x\right)$ is decreasing on the interval $\left(–\infty ,0\right)$ and increasing on the open interval $\left(0,\infty \right).$

Make a table of $f\left(x\right)forx=–2,–1,0,1,2$ and have

$x$	$f(x)=x^{\frac{2}{3}}$
$–2$	$\begin{array}{l}f\left(–2\right)={\left(–2\right)}^{\frac{2}{3}}\\ =1.5874\end{array}$
$–1$	$\begin{array}{l}f\left(–1\right)={\left(–1\right)}^{\frac{2}{3}}\\ =1\end{array}$
$0$	$\begin{array}{l}f\left(0\right)={\left(0\right)}^{\frac{2}{3}}\\ =0\end{array}$
$1$	$\begin{array}{l}f\left(1\right)={\left(1\right)}^{\frac{2}{3}}\\ =1\end{array}$
$2$	$\begin{array}{l}f\left(2\right)={\left(2\right)}^{\frac{2}{3}}\\ =1.5874\end{array}$