Verify that the hypotheses of Rolle’s Theorem are satisfied for f(x) = 16x - x on the interval [0,36], and find the value of c in this interval that satisfies the conclusion of the theorem.

Question

Verify that the hypotheses of Rolle’s Theorem are satisfied for f(x) = 16x - x on the interval [0,36], and find the value of c in this interval that satisfies the conclusion of the theorem.

Jeffrey Jordon · Accepted Answer

Rolle&#039;s Theorem: If a real-valued function f(x) is continuous on a closed interval [a,b], differentiable on the open interval (a, b), and f(a) = f(b), then there is some real number c in the open interval (a, b) such that f (0) = 0f(x)=16x−x [0,36]→f(x) is continuous on [0,36]→f(x) is differentiable on (0,36)→f(0)=f(36)=0Also f′(x)=16−12xf′(0)=16−12xf′(36)=16−1236=112  ⇒ all conditions of Rolle&#039;s Theorem are satisfiedSo answer C=112

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