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Section 3.2

Section 3.2. The Mean Value Theorem. ROLLE’S THEOREM. Theorem : Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval [ a , b ]. 2. f is differentiable on the open interval ( a , b ). 3. f ( a ) = f ( b ).

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Section 3.2

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  1. Section 3.2 The Mean Value Theorem

  2. ROLLE’S THEOREM Theorem: Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval [a, b]. 2. f is differentiable on the open interval (a, b). 3. f (a) = f (b). Then there is a number c in (a, b) such that f′(c) = 0.

  3. THE MEAN VALUE THEOREM (MVT) Theorem: Let f be a function that satisfies the following two hypotheses: 1. f is continuous on the closed interval [a, b]. 2. f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that or, equivalently,

  4. TWO THEOREMS Theorem: If f′(x) = 0 for all x in an interval (a,b), then f is a constant on (a, b). Theorem: If f ′(x) = g′(x) for all x in an interval (a, b), then f – g is a constant on (a, b); that is f(x) = g(x) + c,where c is a constant.

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