CalcRef

Applications of Derivatives · Theorems

Rolle's Theorem

f(a)=f(b)⇒∃ c∈(a,b):f′(c)=0f(a) = f(b) \Rightarrow \exists\, c \in (a,b) : f'(c) = 0

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) where f'(c) = 0. This is a special case of MVT.

Conditions. f continuous on [a,b], differentiable on (a,b), and f(a) = f(b).
  • 2 worked examples
  • 15 in Applications of Derivatives

Worked examples

Find c satisfying Rolle's theorem for f(x) = x² - 4x + 3 on [1, 3].
  1. f(1) = 1-4+3 = 0, f(3) = 9-12+3 = 0. So f(1) = f(3) ✓
  2. f'(x) = 2x - 4 = 0 → x = 2
  3. c = 2 is in (1, 3) ✓

Answer: c = 2

1 more worked examplePremium

Find c satisfying Rolle's theorem for f(x) = sin x on [0, π].

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

Related formulas

Mean Value Theorem

See all 15 Applications of Derivatives formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the 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) where f'(c) = 0. This is a special case of MVT. It is one of the applications of derivatives formulas in the CalcRef reference.

When does the Rolle's Theorem apply?

The Rolle's Theorem holds under this condition: f continuous on [a,b], differentiable on (a,b), and f(a) = f(b).

How do you use the Rolle's Theorem?

Worked example. Find c satisfying Rolle's theorem for f(x) = x² - 4x + 3 on [1, 3]. f(1) = 1-4+3 = 0, f(3) = 9-12+3 = 0. So f(1) = f(3) ✓. f'(x) = 2x - 4 = 0 → x = 2. c = 2 is in (1, 3) ✓. Answer: c = 2.

Practice

Quiz yourself on Applications of Derivatives

Rolle's Theorem is one of 15 Applications of Derivatives formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 15 Applications of Derivatives formulas