`f(x) = cos(2x), [(pi/8), ((7pi)/8)]` 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.
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Given the function `f(x)=cos(2x)` in the interval `[pi/8, (7pi)/8]`
We have to see whether it sattisfies the three hypotheses of Roll's theorem.
(a) f(x) is continuous in the given interval because all periodic functions are continuous.
(b) Now f'(x)=-2sin(2x) which is differentiable everywhere the given interval.
(c) Now evaluate whether `f(pi/8)=f((7pi)/8)`
So there exists a number c such that ` pi/8<c<(7pi)/8`
sin(2c)=0 implies 2c=npi i.e. c=npi/2 , n=1,2,...
So c=pi/2 , pi, 3pi/2 lie in the interval [pi/8, 7pi/8]
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