Question 1

Given the function $$g(x) = x^3$ + 12x^2$ - 15$$, what are the critical points of $$g(x)$$?

Accepted Answer

$$x = 0$$ and $$x = -8$$

Question 2

For the function $$g(x) = x^3$ + 12x^2$ - 15$$, on which interval is the function decreasing?

Accepted Answer

$$(-8, 0)$$

Question 3

If $$g(x) = -x^3$ + 15x^2$, what is g$$'(x)$$?

Accepted Answer

$$-3x^2 + 30x$

Question 4

For $$g(x) = -x^3$ + 15x^2$, at which x$-values does the function have local extrema?

Accepted Answer

$$x = 0$$ and $$x = 10$$

Question 5

What is the value of $$g(10)$$ for $$g(x) = -x^3$ + 15x^2$?

Accepted Answer

$$500$$

Question 6

A function $$f(x)$$ has a local minimum at $$x = a if$$:

Accepted Answer

$$f'(x)$$ changes from negative to positive at $$x = a$$

Question 7

If $$f'(x)$$ changes from positive to negative at $$x = b$$, what does $$f(x)$$ have at $$x = b$$?

Accepted Answer

A local maximum

Question 8

Given $$g(x) = x^3$ + 12x^2$ - 15$$, what is $$g'(x)$$?

Accepted Answer

$$3x^2 + 24x$

Question 9

For $$g(x) = -x^3$ + 15x^2$, what is the nature of the critical point at x = 0$?

Accepted Answer

Local minimum

Question 10

If a function's first derivative is positive on an interval, what can you conclude about the function on that interval?

Accepted Answer

The function is increasing

Question 11

Suppose $$g(x) = x^3$ + 12x^2$ - 15. $$What is the sign of $$g'(x)$$ for $$x \u003e 0$$?

Accepted Answer

Positive

Question 12

Which of the following best describes the process for finding local maxima and minima using derivatives?

Accepted Answer

Find where $$f'(x) = 0$$ and analyze the sign changes of $$f'(x)$$

Question 13

If $$g(x) = -x^3$ + 15x^2$, what is the local maximum value?

Accepted Answer

$$500$$

Question 14

For $$g(x) = x^3$ + 12x^2$ - 15$$, what is $$g'(0)$$?

Accepted Answer

$$0$$

Question 15

If $$g(x) = -x^3$ + 15x^2$, what is g$$'(10)$$?

Accepted Answer

$$0$$

Exam 3 Review: Derivatives, Increasing/Decreasing Functions, and Local Extrema

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