Question

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.) f ( x ) = cos 3 x , [ π / 12 , 7 π / 12 ]

Answer 1

The number c=
`(\pi)/(2)` satisfies conclusion of Roller's theorem.

Explanation:

Given function is,

`f(x)=\cos 3x` which is,

(1) continuous on the closed interval
`\Big[(\pi)/(12),(7\pi)/(12)\Big]`. Since,

`\lim_{x\to (\pi)/(12)}f(x)=(1)/(√(2))=f((\pi)/(12))` and,

`\lim_limits_{x\to (7\pi)/(12)}f(x)=(1)/(√(2))=f((7\pi)/(12))`

(2) derivable in the open interval
`\Big[(\pi)/(12),(7\pi)/(12)\Big]` because of continuity.

(3)
`f((\pi)/(12))=(1)/(√(2))=f((7\pi)/(12))`

Hence all conditions of Rollr's theorem satisfied, so there exist at least one value c, where
`(\pi)/(12)<c<(7\pi)/(12)` such that,

`f'(c)=0\implies -3\sin 3c=0\implies 3c=n\pi\impliesc=(\pi)/(3)n` where n is an integer.

When,

n=1, c=
`(\pi)/(3)\in((\pi)/(12),(7\pi)/(12))`

n=2, c=
`(2\pi)/(3)\\otin((\pi)/(12),(7\pi)/(12))`

Similarly for other values of n, c lies outside of the given interval.

Hence c=
`(\pi)/(2)` satisfies conclusion of Roller's theorem.