Finding Inflection Points

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

Solution We follow the steps for finding an inflection point.

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 \(f.\)