Chapter 4.3, Problem 17E

To determine

To calculate : The points of inflection of the function $y=x^{\frac{1}{3}}\left(x-4\right)$ .

Answer to Problem 17E

The points of inflection of the function $y=x^{\frac{1}{3}}\left(x-4\right)$ are $\left(0,0\right)$ and $\left(-2,{\left(-2\right)}^{\frac{1}{3}}\left(-6\right)\right)$ .

Explanation of Solution

Given information : The function $y=x^{\frac{1}{3}}\left(x-4\right)$ .

Formula used : The point of inflection of the graph is the point where the function has a tangent line an where the concavity of the graph changes.

Use the power, product rule to differentiate.

Calculation :

Consider the function $y=x^{\frac{1}{3}}\left(x-4\right)$ .

Calculate the first derivative of the function using product and power rule.

  $\begin{array}{c}{y}^{\prime }=x^{\frac{1}{3}}\frac{d}{dx}\left(x-4\right)+\left(x-4\right)\frac{d}{dx}{x}^{\frac{1}{3}}\\ =x^{\frac{1}{3}}\left(1\right)+\left(x-4\right)\left(\frac{1}{3}{x}^{-\frac{2}{3}}\right)\\ =x^{\frac{1}{3}}-\frac{4}{3}{x}^{\frac{-2}{3}}+\frac{1}{3}{x}^{\frac{1}{3}}\\ =\frac{4}{3}{x}^{\frac{1}{3}}-\frac{4}{3x^{\frac{2}{3}}}\end{array}$

Calculate the second derivative of the function using power rule.

  $\begin{array}{c}{y}^{″}=\frac{d}{dx}\left(\frac{4}{3}{x}^{\frac{1}{3}}\right)-\frac{d}{dx}\left(\frac{4}{3x^{\frac{2}{3}}}\right)\\ =\frac{4}{3}\left(\frac{1}{3x^{\frac{2}{3}}}\right)-\frac{4}{3}\left(-\frac{2}{3x^{\frac{5}{3}}}\right)\\ =\frac{4}{9x^{\frac{2}{3}}}+\frac{8}{9x^{\frac{5}{3}}}\end{array}$

Set $y^{″}=0$ and solve the equation.

  $\begin{array}{c}\frac{4}{9x^{\frac{2}{3}}}+\frac{8}{9x^{\frac{5}{3}}}=0\\ \frac{4}{9x^{\frac{2}{3}}}=-\frac{8}{9x^{\frac{5}{3}}}\\ -2=x\end{array}$

The inflection points are $\left(0,f\left(0\right)\right)$ and $\left(-2,f\left(-2\right)\right)$ .

Now,

  $\begin{array}{c}f\left(0\right)=y\left(0\right)\\ ={\left(0\right)}^{\frac{1}{3}}\left(0-4\right)\\ =0\end{array}$

And,

  $\begin{array}{c}f\left(-2\right)=y\left(-2\right)\\ ={\left(-2\right)}^{\frac{1}{3}}\left(-2-4\right)\\ ={\left(-2\right)}^{\frac{1}{3}}\left(-6\right)\end{array}$

Draw the graph of the function $y=x^{\frac{1}{3}}\left(x-4\right)$ .

  

It can also be seen from the graph that the concavity of the graph changes at the point $\left(0,0\right)$ and $\left(-2,{\left(-2\right)}^{\frac{1}{3}}\left(-6\right)\right)$ .

Thus, the points of inflection of the function $y=x^{\frac{1}{3}}\left(x-4\right)$ are $\left(0,0\right)$ and $\left(-2,{\left(-2\right)}^{\frac{1}{3}}\left(-6\right)\right)$ .