Calculus I · Unit 2B · lesson

Rolle's Theorem

Concept

Learning objectives

Verify hypotheses and apply Rolle's Theorem.

Equal Endpoints Force a Horizontal Tangent

Explanation

Before the formulas

The theorem in Rolle's Theorem connects global information on an interval with local derivative behavior inside it. Read the hypotheses and conclusion separately. The theorem guarantees existence of at least one point; it may not identify the point, make it unique, or place it at the midpoint.

Use a diagram to understand the claim, then return to algebra to find candidate values when the problem asks for them. A correct theorem citation should name the interval and explain why each hypothesis is satisfied.

Explanation

Equal endpoint heights force at least one horizontal tangent

Rolle's Theorem describes a smooth trip that begins and ends at the same height. Somewhere between, the function must stop rising and begin falling, stop falling and begin rising, or remain level; in every case a horizontal tangent occurs.

The hypotheses matter. A jump, corner, or missing endpoint can defeat the conclusion. Checking continuity and differentiability is part of applying the theorem, not bureaucratic decoration.

Rolle's Theorem formalizes a geometric inevitability: if a smooth curve begins and ends at the same height, then somewhere between those endpoints it must have a horizontal tangent. The theorem does not locate the point; it guarantees one exists.

Every hypothesis has a job. Continuity prevents a jump over the conclusion, differentiability prevents a sharp turn, and equal endpoint values create the horizontal average slope that the derivative must match.

Theorem

Rolle's Theorem

If $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and

f(a)=f(b),

then there exists at least one $c\in(a,b)$ such that

\boxed{f'(c)=0}.

Concept

A smooth trip that begins and ends at the same height must have at least one interior moment with horizontal tangent. The theorem guarantees existence; it does not say the point is unique.

Guided walkthrough

Find Rolle points

Verify Rolle's Theorem for

f(x)=x^2-4x+3

on $[1,3]$ , and find $c$ .

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Common mistake

A theorem conclusion cannot be used until its hypotheses are checked. A function with a corner or discontinuity may have equal endpoint values but fail the differentiability or continuity requirement.

After the explanation

Use the section idea

Reading lens

Turn derivative signs and theorem hypotheses into a defensible account of extrema, monotonicity, concavity, and global shape.

Mental model

Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.

Decision

List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.

Common trap

Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.

Check yourself

Can every turn, bend, endpoint result, and asymptote in your sketch be traced to algebraic evidence?

Source & rights

Original instruction with traceable references.

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