Chapter 2.3, Problem 73AYU

77. $F\left(x\right)=-x^4+8x^2+9$

(a) Determine whether F is even, odd, or neither.

(b) There is a local maximum value of 25 at $x=2$. Determine a second local maximum value.

(c) Suppose the area under the graph of F between $x=0$ and $x=3$ that is bounded from below by the $x\text{-axis}$ axis is 50.4 square units. Using the result from part (a), determine the area under the graph of F between $x=-3$ and $x=0$ that is bounded from below by the $x\text{-axis}$.

To determine

a. Check whether the function $f\left(x\right)=-x^4+8x^2+9$ is even, odd or neither.

Answer to Problem 73AYU

a. The given function $f$ is an even function.

Explanation of Solution

Given:

The function $f\left(x\right)=-x^4+8x^2+9$.

Calculation:

It is asked to check the whether the function $g$ is even, odd or neither and find the second local maximum value based on the local maximum value of 25 at $x=2$. Also find the area under the graph of $F$ between $x=-3$ and $x=0$ that is bounded from below by the $x\text{-axis}$ using the area under the graph of $F$ between $x=0$ and $x=3$ that is bounded from below by the $x\text{-axis}$ is $50.4$ square units.

By the definition of odd and even function,

“A function $f$ is even if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f\left(-x\right)=f\left(x\right)$” and

“A function $f$ is odd if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f\left(-x\right)=-f\left(x\right)$”.

“A function $f$ is odd if and only if, whenever the point $\left(x,y\right)$ is on the graph of $f$, the point $\left(-x,y\right)$ is also on the graph.

a. Consider the function,

$f\left(x\right)=-x^4+8x^2+9$

Replace $x$ by $-x$,

$f\left(-x\right)=-{\left(-x\right)}^4+8{(-x)}^2+9=-x^4+8x^2+9=f\left(x\right)$

From the statement, it can be concluded that the given function $f$ is an even function.

To determine

b. The second local maximum value based on a local maximum given.

Answer to Problem 73AYU

b. The second local maximum value based on the local maximum value of 25 at $x=2$ is $25\text{at}x=2$.

Explanation of Solution

Given:

The function $f\left(x\right)=-x^4+8x^2+9$.

Calculation:

It is asked to check the whether the function $g$ is even, odd or neither and find the second local maximum value based on the local maximum value of 25 at $x=2$. Also find the area under the graph of $F$ between $x=-3$ and $x=0$ that is bounded from below by the $x\text{-axis}$ using the area under the graph of $F$ between $x=0$ and $x=3$ that is bounded from below by the $x\text{-axis}$ is $50.4$ square units.

By the definition of odd and even function,

“A function $f$ is even if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f\left(-x\right)=f\left(x\right)$” and

“A function $f$ is odd if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f\left(-x\right)=-f\left(x\right)$”.

“A function $f$ is odd if and only if, whenever the point $\left(x,y\right)$ is on the graph of $f$, the point $\left(-x,y\right)$ is also on the graph.

b. There is a local maximum value of 25 at $x=2$. Therefore, the local maximum point is $\left(2,25\right)$.

The definition of even function says that whenever the point $\left(2,25\right)$ is on the graph of $f$, the point $\left(-2,25\right)$ also on the graph.

From this, the second local maximum point must be $\left(-2,25\right)$.

To determine

c. The area under the graph of $F$ between $x=-3$ and $x=0$ that is bounded from below by the $x\text{-axis}$ using the area under the graph of $F$ between $x=0$ and $x=3$ that is bounded from below by the $x\text{-axis}$ is $50.4$ square units.

Answer to Problem 73AYU

c. The area under the graph of $F$ between $x=-3$ and $x=0$ that is bounded from below by the $x\text{-axis}$ is $50.4\text{ square units}$.

Explanation of Solution

Given:

The function $f\left(x\right)=-x^4+8x^2+9$.

Calculation:

It is asked to check the whether the function $g$ is even, odd or neither and find the second local maximum value based on the local maximum value of 25 at $x=2$. Also find the area under the graph of $F$ between $x=-3$ and $x=0$ that is bounded from below by the $x\text{-axis}$ using the area under the graph of $F$ between $x=0$ and $x=3$ that is bounded from below by the $x\text{-axis}$ is $50.4$ square units.

By the definition of odd and even function,

“A function $f$ is even if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f\left(-x\right)=f\left(x\right)$” and

“A function $f$ is odd if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f\left(-x\right)=-f\left(x\right)$”.

“A function $f$ is odd if and only if, whenever the point $\left(x,y\right)$ is on the graph of $f$, the point $\left(-x,y\right)$ is also on the graph.

c. The area under the graph of $F$ between $x=0$ and $x=3$ that is bounded from below by the $x\text{-axis}$ is $50.4$ square units.

As the function is even from result (a), there must be another same area under the same graph between $x=0$ and $x=\text{−}3$ that is also bounded from below by the $x\text{-axis}$.

Therefore the area under the graph of $F$ between $x=\text{−}3$ and $x=0$ that is bounded from below by the $x\text{-axis}$ $50.4$ square units.