5.1 Using the Mean Value Theorem

What the Mean Value Theorem Says

If a function f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) − f(a)) / (b − a). This means the instantaneous rate of change at c equals the average rate of change over the interval.

Checking Hypotheses First

Before applying MVT, always verify continuity on [a, b] and differentiability on (a, b). If either condition fails (e.g., a corner, discontinuity, or vertical tangent), MVT may not guarantee a valid c exists. Piecewise functions and absolute value functions are common places where differentiability fails.

MVT vs. Rolle's Theorem

Rolle's Theorem is a special case of MVT where f(a) = f(b), guaranteeing a point c where f'(c) = 0. MVT generalizes this to any secant line slope. Both theorems only guarantee existence of c — they don't tell you how many such points exist or their exact value without solving.

Key terms

Mean Value Theorem (statement)

If f is continuous on [a, b] and differentiable on (a, b), there exists c in (a, b) such that f'(c) = (f(b) − f(a)) / (b − a).

MVT hypotheses

f must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b).

Rolle's Theorem

A special case of MVT: if f(a) = f(b), there exists c in (a, b) where f'(c) = 0.

Geometric meaning of MVT

There is a point c where the tangent line's slope equals the slope of the secant line connecting (a, f(a)) and (b, f(b)).

When MVT fails

If f has a discontinuity or a non-differentiable point (like a corner or vertical tangent) on [a, b], MVT cannot guarantee such a c exists.

Try an AP-style question

For f(x) = x² on [1, 3], what value c satisfies the Mean Value Theorem?

1. 2
2. 4
3. 1.5
4. √5