Mean Value Theorem Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Mean Value Theorem.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one point c in (a, b) where f'(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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The MVT connects a function's average rate of change over an interval to its instantaneous rate at some interior point. Geometrically, there's a point where the tangent line is parallel to the secant line through the endpoints.

Common stuck point: The MVT is an existence theorem—it guarantees some c exists but doesn't specify which one (or how many). Finding c explicitly is a computation exercise, but the theorem's power is in the guarantee.

Sense of Study hint: Compute the average slope (f(b)-f(a))/(b-a) first, then set f'(x) equal to it and solve for x in (a,b).

Worked Examples

Example 1

easy
Verify the MVT for f(x) = x^2 on [1, 3] by finding the value of c.

Solution

  1. 1
    Check hypotheses: f is a polynomial, so continuous on [1,3] and differentiable on (1,3). ✓
  2. 2
    Average rate of change: \frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4.
  3. 3
    Find c: f'(x) = 2x. Set 2c = 4 \Rightarrow c = 2.
  4. 4
    Check: c = 2 \in (1, 3). ✓

Answer

c = 2 \in (1, 3), confirming the MVT.
The MVT guarantees a c where the instantaneous rate equals the average rate. Here c=2 is where the tangent to y = x^2 is parallel to the secant through (1,1) and (3,9).

Example 2

hard
Use the MVT to prove: if f'(x) = 0 for all x in (a, b), then f is constant on (a, b).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the value of c guaranteed by the MVT for f(x) = x^3 on [0, 2].

Example 2

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

These ideas may be useful before you work through the harder examples.

derivativelimitintermediate value theorem