Analyzing Functions Test – AP Calculus AB Math (Printable PDF)

Question 1 Absolute maximum (2 points)

What is the absolute maximum value of f(x) = x³ − 12x on [−3, 5]?

1. 16
2. 65
3. 9
4. 5

Question 2 Local maximum from a graph (2 points)

The graph of f′(x) is shown. At which x-value does f have a local maximum?

1. −2
2. 0
3. 1
4. There is none

Question 3 Mean Value Theorem application (3 points)

Let f(x) = x³ − 2x on [0, 3].

1. Verify that the hypotheses of the Mean Value Theorem are satisfied. (1 point)
2. Find all values c that satisfy the conclusion. (2 points)

Question 4 Concavity analysis (3 points)

Let g(x) = x⁴ − 8x³ + 18x².

1. Find g″(x). (1 point)
2. Find the intervals where g is concave up and concave down. (1 point)
3. Find the inflection points and justify. (1 point)

Question 5 Minimizing the surface of a box (3 points)

A closed box with a square base has volume 1,728 cm³. Let s be the base side.

1. Write the surface area S as a function of s alone. (1 point)
2. Determine the base side that makes S smallest. (1 point)
3. Justify that it is a minimum. (1 point)

Question 6 First derivative test (3 points)

Let f(x) = 2x³ − 9x² + 12x − 1.

1. Find f′(x). (1 point)
2. Use a sign chart for f′(x) to classify each critical number. (1 point)
3. Find the local maximum and minimum values. (1 point)

Question 7 Analyzing a graph (2 points)

The graph of f is shown. Give the interval on which f is increasing and the interval on which f is concave up.

Question 8 Extreme Value Theorem hypotheses (2 points)

Does the Extreme Value Theorem guarantee an absolute maximum for g(x) = 1/(x − 1) on [0, 2]? Explain, and state whether g actually has an absolute maximum.