`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...

`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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Chapter 4, 4.2 - Problem 4 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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nees101 | Student, Graduate | (Level 2) Adjunct Educator

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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,`````````f(pi/8)=cos(2.pi/8)=cos(pi/4)=1/sqrt(2)```

`f((7pi)/8)=cos(2.(7pi)/8)=cos((7pi)/4)=1/sqrt(2)`

Therefore `f(pi/8)=f((7pi)/8)`

So there exists a number c such that ` pi/8<c<(7pi)/8`

and f'(c)=0

Therefore, f'(c)=-2sin(2c)=0

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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