Rolle's theorem
f(x)=sin2x from [0, pi/2]
determine whether rolle's theorem applies, if so find the point(s) that are guaranteed to exist by the theorem

Rolles Theorem applies to differentiable functions that have equal values at two different points. The theorem states that in this case there is an intermediate point wich is stationaty, this is its derivative is zero.

Here,
sin(2x), x = 0 => sin(0) = 0 and x = pi/2 =>sin(2*pi/2) = sin (pi) = 0.

Then, by the theorem there is a point between x=0 and x = pi/2 where the derivative of sin (2x) = 0.

if you reduce the interval step by step you could find where the derivative is zero.

For example x = pi/8 and x = 3pi/8

sin(2pi/8) = sin (pi/4) = √2/2
sin (2*3pi/8) = sin (3pi /4) = √2/2

Then there is a stationary point between pi/8 and 3pi/8

You can find the point derivating the function [sin(2x)] ' = 2 cos(2x) = 0

cos(2x) = 0 => 2x = pi/2 => x = pi/4, which is the point searched.