Question

Directions: Given the following information, find and classily any relative extrema of / (x). Justify your answers.

Answer 1

We have to find the extrem values (minimum and maximum) of f(x):

$f(x)=x^2e^x$

We can find the extreme values by deriving f(x) and equal it to 0 to find the values of x for the extreme points.

To derive f(x) we have to apply the multiplication rule:

$\begin{gathered} f(x)=g(x)\cdot h(x) \\ \Rightarrow f^{\prime}(x)=g^{\prime}(x)\cdot h^{}(x)+g(x)\cdot h^{\prime}(x) \end{gathered}$

Applied to f(x), we get:

$\begin{gathered} f^{\prime}(x)=2x\cdot e^x+x^2e^x \\ f^{\prime}(x)=x(x+2)e^x \end{gathered}$

If we equal this to 0 we get:

$\begin{gathered} f^{\prime}(x)=0 \\ x(x+2)e^x=0 \\ x(x+2)=\frac{0}{e^x} \\ x(x+2)=0 \\ x_1=0 \\ x_2+2=0\Rightarrow x_2=-2 \end{gathered}$

Answer: We have two extrema of f(x): one at x = 0 and the other at x = -2.