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

2 Answers
Feb 2, 2018

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.

Feb 10, 2018

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.