Question

How do you determine if rolles theorem can be applied to `f(x) = sin 2x` on the interval [0, (pi/2)] and if so how do you find all the values of c in the interval for which f'(c)=0?

Answer 1

you do this by determining whether the hypothses for Rolle's Theorem (the "If" parts) are true for `f(x) = sin 2x` on the interval [0, (pi/2)]

Hypothesis 1: we want: `f` is continuous on the closed interval `[a,b]`

So, is `sin2x` continuous on `[0, pi/2]`? Why should the reader agree with your answer?

Hypothesis 2: we want: `f` is differentiable on the open interval `(a,b)`

So, is `sin2x` differentiable on `(0, pi/2)`? Why should the reader agree with your answer?

Hypothesis 3: we want `f(a) = f(b)`

So, is `sin(2(0)) = sin (2( pi/2))`? Why should the reader agree with you?

To find all values of `c` in the interval for which `f'(c) = 0`, find `f'(x)`, set it equal to `0` and solve the equation, ignoring solutions outside the interval.