QuestionDetermine if $g(x)=x^{3}-48 x$ is even, odd, or neither. Find the local maximum value given a local minimum of -128 at 4.

Studdy Solution

STEP 1

Assumptions1. The function is $g(x)=x^{3}-48x$ . We are asked to determine whether the function is even, odd, or neither.
3. We are also asked to find the local maximum value given that there is a local minimum value of -128 at x=4.

STEP 2

To determine whether a function is even, odd, or neither, we need to evaluate $g(-x)$ and compare it with $g(x)$ .
The function is even if $g(-x) = g(x)$ , odd if $g(-x) = -g(x)$ , and neither if it doesn't satisfy either of these conditions.
So, let's calculate $g(-x)$ .
$g(-x) = (-x)^{} -48(-x)$

STEP 3

implify the expression.
$g(-x) = -x^{3} +48x$

STEP 4

Now, compare $g(-x)$ with $g(x)$ .We see that $g(-x) \neq g(x)$ and $g(-x) \neq -g(x)$ , so the function is neither even nor odd.

STEP 5

Now, let's find the local maximum value.Since we know that there is a local minimum at $x=4$ , we can use the symmetry of the cubic function to find the local maximum.The cubic function $g(x)=x^{3}-48x$ is symmetric about the line $x=0$ .

STEP 6

Since the function is symmetric about $x=0$ , the local maximum will occur at the same distance from $x=0$ as the local minimum, but on the opposite side.So, the local maximum will occur at $x=-4$ .

STEP 7

Substitute $x=-4$ into the function $g(x)$ to find the local maximum value.
$g(-4) = (-4)^{3} -48(-4)$

STEP 8

Calculate the value.
$g(-4) = -64 +192 =128$ So, the local maximum value of the function is128.

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