Identify the open intervals on which the function is increasing or decreasing. (Enter your answer using interval notation.)

increasing
decreasing

Question

Identify the open intervals on which the function is increasing or decreasing. (Enter your answer using interval notation.) $$ g(x)=x^{2}-4 x-32 $$ increasing decreasing

UpStudy AI · Accepted Answer

We begin by finding the derivative of $$ g(x) = x^2 - 4x - 32 $$:

$$
g'(x) = 2x - 4.
$$

Set the derivative equal to 0 to find the critical point:

$$
2x - 4 = 0 \quad \Rightarrow \quad x = 2.
$$

For values of $$ x $$ less than 2, choose a test point (e.g., $$ x = 0 $$):

$$
g'(0) = 2(0) - 4 = -4 \quad (\text{negative}).
$$

Thus, $$ g(x) $$ is decreasing on the interval:

$$
(-\infty, 2).
$$

For values of $$ x $$ greater than 2, choose a test point (e.g., $$ x = 3 $$):

$$
g'(3) = 2(3) - 4 = 2 \quad (\text{positive}).
$$

Thus, $$ g(x) $$ is increasing on the interval:

$$
(2, \infty).
$$
The function $$ g(x) = x^2 - 4x - 32 $$ is decreasing on $$ (-\infty, 2) $$ and increasing on $$ (2, \infty) $$.

Identify the open intervals on which the function is increasing or decreasing. (Enter your answer using interval notation.)

increasing
decreasing

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Identify the open intervals on which the function is increasing or decreasing. (Enter your answer using interval notation.) increasing decreasing

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