Chapter 4.3, Problem 13E

To determine

To calculate : The points of inflection of the function $y=x{e}^x$ .

Answer to Problem 13E

The points of inflection of the function $y=x{e}^x$ is $\left(-2,\frac{-2}{e^2}\right)$ .

Explanation of Solution

Given information : The function $y=x{e}^x$ .

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 product and power rule to differentiate.

Calculation :

Consider the function $y=x{e}^x$ ,

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

  $y^{\prime }=x{e}^x+e^x$

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

  $\begin{array}{c}{y}^{″}=x{e}^x+e^x+e^x\\ =x{e}^x+2e^x\end{array}$

Set $y^{″}=0$ .

  $\begin{array}{c}x{e}^x+2e^x=0\\ e^x\left(x+2\right)=0\end{array}$

The value of exponential is never 0. The function $y^{″}$ will only be $0$ if $x+2=0$ that is $x=-2$ .

The factor $\left(x+2\right)$ changes sign at the point $x=-2$ . Then $y^{″}$ must also change the sign at this number.

Then the points of inflection are $\left(-2,f\left(-2\right)\right)$ that is $\left(-2,\frac{-2}{e^2}\right)$ .

The solution can be confirmed from the graph as follows.

  

Thus, the points of inflection of the function $y=x{e}^x$ is $\left(-2,\frac{-2}{e^2}\right)$ .