Question

\(G(x)=-x^{4}+32 x^{2}+144\) (a) Determine whether \(G\) is even, odd, or neither. (b) There is a local maximum value of 400 at \(x=4\). Find a second local maximum value. (c) Suppose the area of the region enclosed by the graph of \(G\) and the \(x\) -axis between \(x=0\) and \(x=6\) is 1612.8 square units. Using the result from (a), determine the area of the region enclosed by the graph of \(G\) and the \(x\) -axis between \(x=-6\) and \(x=0\).

Short answer

(a) The function is even. (b) The second local maximum value is 400. (c) The area is 1612.8 square units.

Step-by-step solution

- Determine if the function is even, odd, or neither

To examine the symmetry of the function, compute G(-x) and compare it to the original function G(x). If G(-x) = G(x), the function is even. If G(-x) = -G(x), the function is odd. Otherwise, it is neither. Compute G(-x): G(-x) = -(-x)^4 + 32(-x)^2 + 144 Simplify: G(-x) = -x^4 + 32x^2 + 144 Comparison: G(-x) = G(x) thus the function is even.

- Find the second local maximum value

Given that there is a local maximum value of 400 at x=4, use the symmetry of the even function to determine the second local maximum value. Since the function is even, G(x) = G(-x). Hence, there is also a local maximum at x = -4. Since G(4) = 400, G(-4) = 400. Thus, the second local maximum value is 400.

- Calculate area as per symmetry

Given that the area between x = 0 and x = 6 is 1612.8 square units, utilize the even property of the function to determine the area between x = -6 and x = 0. Since G(x) is even, the area between x = -6 and x = 0 is equal to the area between x = 0 and x = 6. Therefore, the area between x = -6 and x = 0 is also 1612.8 square units.

Key concepts

Even and Odd Functions

Understanding whether a function is even, odd, or neither can greatly simplify solving problems.

An **even function** is symmetric about the y-axis. This means that if you fold the graph along the y-axis, both halves will match perfectly.
You can test for an even function by examining its equation: if replacing \( x \) with \( -x \) results in the same function ( \( G(-x) = G(x) \) ), then the function is even.

An **odd function** is symmetric about the origin. This symmetry implies that if you rotate the graph 180 degrees around the origin, it will look the same. To test for an odd function, check if \( G(-x) = -G(x) \).
In our exercise, for the function \(G(x) = -x^{4} + 32x^{2} + 144 \), calculating \( G(-x) \) gives us
\[ G(-x) = -(-x)^{4} + 32(-x)^{2} + 144 \]

Simplifying this, we get
\[ G(-x) = -x^{4} + 32x^{2} + 144 \]

Notice that \( G(-x) = G(x) \), so the function is even.

Local Maximum

A local maximum is a point where the function attains a peak value in a particular interval.

In other words, it is higher than any other value in its immediate vicinity. To find these points, we often use calculus (finding derivatives and solving for critical points).

Once we know the function is even, it helps in easily identifying other local maxima. For the function \[ G(x) = -x^{4} + 32x^{2} + 144 \], it has a local maximum value of 400 at \( x = 4 \). Because the function is even, it is symmetric around the y-axis. This means that there is another local maximum with the same value at \( x = -4 \).

Therefore, we have two local maxima at \( x = 4 \) and \( x = -4 \), both with the value 400.

Area Under the Curve

The area under the curve of a function gives us the total value between the curve and the x-axis over a specified interval.

This is a critical concept in integral calculus and can be used in various applications such as physics, engineering, and economics.

For our function \[ G(x) = -x^{4} + 32x^{2} + 144 \], finding the area between \( x = 0 \) and \( x = 6 \) gives us 1612.8 square units.

Because the function is even, we know its graph is symmetric about the y-axis. This means the area between \( x = -6 \) and \( x = 0 \) is identical to the area between \( x = 0 \) and \( x = 6 \).

Therefore, the total area for the interval from \( x = -6 \) to \( x = 0 \) is also 1612.8 square units.

Using the symmetry of even functions can greatly simplify calculations involving areas.