Where is Rolle's Theorem true?

Question

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

Charles Arocha · Accepted Answer

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

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#.

Isaiah Allen · Answer

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.

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.

Aurora Alvarado · Answer

undefined Rolle's Theorem is true for any continuous function $ f(x) $ that is differentiable on the closed interval $[a, b]$, where $ f(a) = f(b) $. In other words, if a function is continuous on a closed interval and differentiable on the open interval, and it takes the same values at the endpoints of the interval, then there exists at least one point $ c $ in the open interval $(a, b)$ such that the derivative of the function evaluated at $ c $ is equal to zero.