Practice Mean Value Theorem in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

If ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there exists at least one point cc in (a,b)(a, b) where f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}

If you drive 150 miles in 2 hours, your average speed is 75 mph. The MVT says at some instant during the trip, your speedometer read exactly 75 mph. The instantaneous rate must equal the average rate at least once.

Showing a random 20 of 50 problems.

Example 1

medium
Two functions satisfy f′(x)=g′(x)f'(x)=g'(x) on an interval. What does the MVT-based corollary say?

Example 2

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A car travels 120 miles in 2 hours. Explain why the MVT guarantees the car exceeded 60 mph at some instant.

Example 3

challenge
Two differentiable functions ff and gg satisfy f(0)=g(0)f(0)=g(0) and f′(x)<g′(x)f'(x)<g'(x) for all x>0x>0. Prove f(x)<g(x)f(x)<g(x) for x>0x>0.

Example 4

medium
For f(x)=1xf(x)=\frac{1}{x} on [1,3][1,3], find cc from the MVT.

Example 5

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Use the MVT to show ∣sin⁡a−sin⁡b∣≤∣a−b∣|\sin a-\sin b|\le|a-b|.

Example 6

medium
A function satisfies f(2)=5f(2)=5 and f′(x)≤3f'(x)\le 3 for all xx. Find an upper bound for f(6)f(6).

Example 7

easy
State the two hypotheses required for the Mean Value Theorem.

Example 8

medium
Find cc from the MVT for f(x)=x3f(x)=x^3 on [0,2][0,2].

Example 9

challenge
Show that the equation x3+x−1=0x^3+x-1=0 has exactly one real root using MVT (or its monotonicity corollary).

Example 10

medium
Why does f(x)=x2/3f(x)=x^{2/3} fail MVT hypotheses on [−1,1][-1,1]?

Example 11

easy
What is the geometric interpretation of the MVT?

Example 12

easy
For f(x)=2x+1f(x)=2x+1 on [0,3][0,3], what does the MVT give for f′(c)f'(c)?

Example 13

easy
Driving 120120 miles in 22 hours, the MVT guarantees what instantaneous speed occurred?

Example 14

hard
Prove using MVT: if f′(x)>0f'(x)>0 on (a,b)(a,b), then ff is strictly increasing on (a,b)(a,b).

Example 15

hard
Suppose ff is differentiable with f(1)=2f(1)=2 and f′(x)≥4f'(x)\ge 4 on [1,5][1,5]. Find a lower bound for f(5)f(5).

Example 16

medium
Find cc from MVT for f(x)=sin⁡xf(x)=\sin x on [0,π][0,\pi].

Example 17

easy
Find the average rate of change of f(x)=x2f(x)=x^2 on [2,5][2,5].

Example 18

easy
For f(x)=x2f(x)=x^2 on [1,4][1,4], find the cc guaranteed by MVT.

Example 19

medium
For f(x)=1xf(x)=\frac{1}{x} on [1,2][1,2], find the MVT point cc.

Example 20

easy
For f(x)=3x+2f(x)=3x+2 on [0,5][0,5], what is f′(c)f'(c) from the MVT?
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