Rolle's Theorem

2021-10-03

Verify that the hypotheses of Rolle’s Theorem are satisfied for f(x) = \(1\over6 \)x - \(\sqrt {x}\) on the interval [0,36], and find the value of c in this interval that satisfies the conclusion of the theorem.

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Answered 2021-10-20 Author has 1944 answers

Rolle'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) = 0

\(f(x)=\frac{1}{6}x-\sqrt x\) \([0,36]\)

\(→ f(x)\) is continuous on \([0,36]\)
\(→ f(x)\) is differentiable on \((0,36)\)
\(→ f(0) = f(36) = 0\)

Also \(f '(x)= \frac{1}{6}-\frac{1}{2\sqrt x}\)

\(f'(0)=\frac{1}{6}-\frac{1}{2\sqrt x}\)

\(f'(36)=\frac{1}{6}-\frac{1}{2\sqrt 36}=\frac{1}{12}\)

 

 

\(⇒\) all conditions of Rolle's Theorem are satisfied

So answer \(C=\frac{1}{12}\)

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