Rolle’s Theorem and the
Mean Value Theorem
Guaranteeing Extrema
Rolle’s Theorem
• Guarantees the existence of an extreme
value in the interior of a closed interval
Rolle’s Theorem
• Let f be a continuous function on the
closed interval [a, b] and differentiable on
the open interval (a, b). If
• f(a) = f(b)
• then there is at least one number c in (a,
b) such that f’(c) = 0.
Rolle’s Theorem
• Three things that must be true for the
theorem to hold:
• (a) the function must be continuous
• (b) the function must be differentiable
• (c) f(a) must equal f(b)
Rolle’s Theorem Examples
• Book
• On-line
Mean Value Theorem
• If f is continuous on the closed interval
• [a, b] and differentiable on the open interval
(a, b), then there exists a number c in (a, b)
such that
f (b)  f (a)
f '(c) 
ba
Mean Value Theorem
• In other words, the derivative equals the
slope of the line.
Mean Value Theorem
• Tutorial
• Examples
• More Examples