Question

Where is Rolle's Theorem true?

f(x) = 2 tan(x/2) find the point in the interval [0, 2pi] where the conclusion of Rolle's Theorem is true

Answer 1

Given `f(x)=2tan(x/2)`, Since the graph of tangent is not differentiable at `x=pi and 2pi`, Rolle's Theorem does not apply.

Explanation:

Rolle's Theorem applies when `f(x)` is continuous, differentiable, and when `f(a)=f(b)`, so there exists a value `c` such that `f'(c)=0`

Since the graph of tangent is not differentiable at `x=pi and 2pi`, Rolle's Theorem does not apply.

Answer 2

For `f(x) = 2tan(x/2)` there is no point in the interval `[0,2pi]` where the conclusion of Rolle's Theorem is true.

Explanation:

The conclusion of Rolle's Theorem involves solving `f'(x) = 0`.

But for `f(x) = 2tan(x/2)`, we have `f'(x) = sec^2(x/2)` and we know, from trigonometry, that `sec^2(t) > 1` for all real `t`.
Therefore, we cannot solve `f'(x) = 0` for this function, `f`.