Chapter 2.3, Problem 72AYU

(a)

To determine

To find: whether $f$ is even, odd, or neither.

Answer to Problem 72AYU

  $f\left(x\right)$ is an odd function.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=-x^3+12x$

Calculation:

Replacing $x$ by $-x$ in the given function $f\left(x\right)=-x^3+12x$ , we get

  $\begin{array}{c}f\left(-x\right)={\left(-x\right)}^3+12\left(-x\right)\\ =x^3-12x\\ =-f\left(x\right)\end{array}$

Since $f\left(-x\right)=-f\left(x\right)$ , therefore $f\left(x\right)$ is an odd function.

(b)

To determine

To find: the local minimum value.

Answer to Problem 72AYU

The local minimum value is $-16$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=-x^3+12x$

There is a local maximum value of $16$ at 2.

Calculation:

The graph of the function $f\left(x\right)=-x^3+12x$ is shown below:

  

Since the function is odd, therefore the graph is symmetric with respect to the origin.

Also, since the local maximum value is given as 16 at $x=2$ , so using the symmetry of the graph with respect to the origin, the local minimum value will be $-16$ at $x=-2$ .

Therefore, the local minimum value is $-16$ .