Question

Check if the given function satisfies the three hypotheses of the Rolle’s Theorem. If Rolle’sTheorem applies, find all the values that satisfy the conclusion of Rolle’s Theorem. If it does not apply, state why?I hope the picture is clear and I need help

Answer 1

Please see below.

Explanation:

Accordng to Rolle’s Theorem, if `a < b` and

1. if `f` is continuous on the closed interval `[a,b]` and
2. differentiable on the open interval `(a,b)`
3. and `f(a)=f(b)`,

then there is a `c` in `(a, b)` with `f'(c)=0`.

which means that under these hypotheses, `f` has a horizontal tangent somewhere between `a` and `b`.

Here we have `f(x)=tanx` on the closed interval `[0,2pi]`

and although we have `f(0)=f(2pi)=0` on the closed interval `[0,2pi]`,

but neither `f(x)` is continuous on the closed interval `[a,b]`, nor `f(x)=tanx` is differentiable on the open interval `(a,b)`

Hence, we cannot be sure that there is a `c` in `(0, 2pi)` with `f'(c)=0`.

Observe that `f'(x)=sec^2x` and it cannot have a value `0`.