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#.