Chapter 5.1, Problem 14E

To determine

To find: theextreme values of the function on the interval and where they occur by using analytic method and find critical points that are not stationary points.

Answer to Problem 14E

themaximum value is $1$ at $x=0$ and no minimum value.

Explanation of Solution

Given information:

The function is $k\left(x\right)=e^{-x^2},-\infty \le x\le \infty$

Calculation: to find the extreme values of function, first find its derivative,

  $\begin{array}{l}k\left(x\right)=e^{-x^2}\\ k\text{'}\left(x\right)=2x{e}^{-x^2}\text{ [differentiate and simplify]}\end{array}$

At point $x=0$ where $f\text{'}\left(x\right)=0$ so the function does not have any critical points, and the point $x=\infty \text{ and }x=-\infty$ , the value of function does not exist,

Now substitute in function,

  $\begin{array}{l}k\left(x\right)=e^{-x^2}\\ k\left(x\right)=1\text{ [substitute }x=0\text{ and simplify]}\end{array}$

Thus, the maximum value is $1$ at $x=0$ and no minimum value.