Mean Value Theorem

The central idea

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one $c\in(a,b)$ satisfying $f'(c)=[f(b)-f(a)]/(b-a)$. Rolle's theorem is the special case $f(a)=f(b)$, which guarantees $f'(c)=0$.

Definitions, hypotheses, and notation

The two hypotheses cover different possible failures. A jump violates continuity, while a corner or cusp may preserve continuity but violate differentiability. In either case the endpoint secant slope need not occur as an interior tangent slope. Verifying both conditions is therefore mathematical content, not a ceremonial preface.

The theorem underlies many consequences: a function with zero derivative throughout an interval is constant; derivative bounds control total change; and equal derivatives imply functions differ by a constant. It can establish existence of a point but normally not locate it without solving $f'(c)$ equal to the computed average rate.

A dependable method and decision rule

1. State and verify continuity on the closed interval.
2. State and verify differentiability on the open interval.
3. Compute the endpoint secant slope.
4. Set $f'(c)$ equal to that slope and solve.
5. Keep only solutions strictly inside $(a,b)$ and state the conclusion.

Fully worked example

Common mistakes and why they fail

Verify hypotheses before solving for the point

The theorem requires continuity on $[a,b]$ and differentiability on $(a,b)$. Only then solve $f'(c)=[f(b)-f(a)]/(b-a)$. A corner, cusp, jump, or vertical tangent inside the interval can invalidate the conclusion. At least one $c$ is guaranteed, not exactly one; several tangents may be parallel to the secant. Rolle's theorem is the special case $f(a)=f(b)$, where the required slope is zero. Discard algebraic solutions outside $(a,b)$. If the hypotheses hold but no valid solution remains, recheck the derivative or equation solving rather than concluding the theorem failed.