Proof Prove the following generalization of the Mean Value Theorem. If f is twice differentiable on the closed interval , then
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Proof Use the Product Rule twice to prove that if f, g, and h are differentiable functions of x, then
Darboux's Theorem
Prove Darboux’'s Theorem: Let f be differentiable on the closed interval [a, b] such that , and . If d lies between and , then there exists c in (a, b) such that .
Extended Mean Value Theorem
Prove the Extended Mean Value Theorem: If f and are continuous on the closed interval [a, b], and if exists in the open interval (a, b), then there exists a number c in (a, b) such that
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