Question

cos2x 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. (Enter your answers as a comma-separated list.)

Answer 1

Answer with Step-by-step explanation:

We are given that

`f(x)=cos2x`

[
`(\pi)/(8),(7\pi)/(8)`]

1.Cos2x is continuous on given interval [
`(\pi)/(8),(7\pi)/(8)`]

2.Cos 2x is differentiable in (
`(\pi)/(8),(7\pi)/(8)`)

3.
`f((\pi)/(8))=Cos2((\pi)/(8))=cos(\pi)/(4)=(1)/(\sqrt 2)`

`f((7\pi)/(8))=Cos((7\pi)/(4)=Cos(2\pi-(\pi)/(4))=Cos(\pi)/(4)=(1)/(\sqrt 2)`

Using the formula

`Cos(2\pi -x)=Cos x`

Therefore, f(a)=f(b)

Hence,Cos 2x satisfies the three hypothesis of Roll's theorem on the given interval.

`f'(x)=-2Sin2 x`

Substitute x=c

`f'(c)=-2Sin2x`

`f'(c)=0`

`-2Sin2x=0`

`2x=n\pi`

`x=(n\pi)/(2)`

Where
`n\in Z`

Substitute n=1

`x=(\pi)/(2)` lies in the given interval.

Hence, the value of c=
`(\pi)/(2)`