Chapter 2.3, Problem 74AYU

(a)

To determine

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

Answer to Problem 74AYU

  $G\left(x\right)$ is an even function.

Explanation of Solution

Given information:

Given function

  $G\left(x\right)=-x^4+32x^2+144$

Calculation:

Replacing $x$ by $-x$ in the given function $G\left(x\right)=-x^4+32x^2+144$ , we get

  $\begin{array}{c}G\left(x\right)=-{\left(x\right)}^4+32{\left(x\right)}^2+144\\ G\left(-x\right)=-{\left(-x\right)}^4+32{\left(-x\right)}^2+144\\ =-x^4+32x^2+144\\ =G\left(x\right)\end{array}$

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

(b)

To determine

To find: the second local maximum value.

Answer to Problem 74AYU

T he second local maximum value is 400.

Explanation of Solution

Given information:

Given function

  $G\left(x\right)=-x^4+32x^2+144$

There is a local maximum value of 400 at $x=4$ .

Calculation:

The graph of the function $G\left(x\right)=-x^4+32x^2+144$ is shown below:

  

Since the function is even, therefore the graph is symmetric with respect to the $x$ -axis.

Also, since one local maximum value is given as 400 at $x=4$ , so using the symmetry of the graph with respect to the $x$ axis, the second local maximum value will be 400 at $x=-4$ .

Therefore, the second local maximum value is 400.

(c)

To determine

To find: the area under the graph of G between $x=-6$ and $x=0$ bounded below by the $x$ axis.

Answer to Problem 74AYU

The area under the graph of G bounded by the $x$ axis, $x=-6$ ,and $x=0$ is 1612.8 square units.

Explanation of Solution

Given information:

Given function

  $G\left(x\right)=-x^4+32x^2+144$

Suppose the area under the graph of $G$ between $x=0$ and $x=6$ that is bounded below by the $x$ axis is 1612.8 square units.

Calculation:

The area bounded by the graph of G, $x=-6,x=6$ , $x$ axis and y axis is shown below:

  

Since the graph is symmetric with respect to the $x$ axis, so the area under the graph of $F$ bounded by the $x$ axis, $x=-6$ ,and $x=0$ is same as that of area under the graph of $F$ bounded by the $x$ axis, $x=6$ and $x=0$ .

Therefore, the area under the graph of G bounded by the $x$ axis, $x=-6$ ,and $x=0$ is 1612.8 square units.