Figure

EXAMPLE 6Finding Inflection Points

Find the inflection points of $f(x)=x^{5/3}$ .

Solution We follow the steps for finding an inflection point.

$f(x)=x^{5/3}$

Step 1 The domain of $f$ is all real numbers. The first and second derivatives of $f$ are ${f^\prime (x)=\frac{5}{3}x^{2/3}}\qquad f^{\prime \prime} (x)=\frac{10}{9}x^{-1/3}=\frac{10}{9x^{1/3}}$

The second derivative of $f$ does not exist when $x =0$ . So, $(0,0)$ is a possible inflection point.

Step 2 Now use the Test for Concavity.

$\hbox{If }x<0\quad \text{then}\quad$

$f^{\prime\prime} (x)<0$

$\hbox{ so }f\hbox{ is concave down on}\;(-\infty ,0).$

$\hbox{If }x>0\quad \text{then}\quad$

$f^{\prime\prime} (x)>0$

$\hbox{ so }f\hbox{ is concave up on }\;(0,\infty ).$

Step 3 Since the concavity of $f$ changes at $0$ , we conclude that $(0,0)$ is an inflection point of $f.$