# 4.2 - The Mean Value Theorem

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4.2 - The Mean Value Theorem
Theorems If the conditions (hypotheses) of a theorem are satisfied, the conclusion is known to be true. If the hypotheses of a theorem are not satisfied, the conclusion may still be true, but not guaranteed.

Rolle’s Theorem Let f be a function that satisfies the following three hypotheses: f is continuous on the closed interval [a, b]. f is differentiable on the open interval (a, b). f (a) = f (b) Then there is a number c in (a, b) such that f ′(c) = 0.

Examples: Rolle’s Theorem
Explain why the conclusion to Rolle’s Theorem is not guaranteed for the function f (x) = x / (x – 3) on the interval [1, 6]. 2. Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.

The Mean Value Theorem Let f be a function that satisfies the following two hypotheses: f is continuous on the closed interval [a, b]. f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that

Example: Mean Value Theorem
3. Verify that the function satisfies the two hypotheses of Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Mean Value Theorem.

Theorem If f ′(x) = 0 for all x in an interval (a, b), then f is constant on (a, b). Corollary If f ′(x) = g′(x) for all x in an interval (a, b), then f – g is constant on (a, b); that is, f(x) = g(x) + c where c is a constant. Example: If f (x) = x2 + 3 and g(x) = x2 + 7, find f ′(x), g ′(x), and (f – g)(x).