Question

How do you determine if rolles theorem can be applied to `f(x)= 3sin(2x)` on the interval [0, 2pi] and if so how do you find all the values of c in the interval for which f'(c)=0?

Answer 1

The Rolles theorem says that if:

1. `y=f(x)` is a continue function in a set `[a,b]`;
2. `y=f(x)` is a derivable function in a set `(a,b)`;
3. `f(a)=f(b)`;

then at least one `cin(a,b)` as if `f'(c)=0` exists.

So:

1. `y=3sin(2x)` is a function that is continue in all `RR`, and so it is in `[0,2pi]`;
2. `y'=6cos(2x)` is a function continue in all `RR`, so our function is derivable in all `RR`, so it is in `[0,2pi]`;
3. `f(0)=f(2pi)=0`.

To find `c`, we have to solve:

`y'(c)=0rArr6cos(2c)=0rArrcos(2c)=0rArr`

`2c=pi/2+2kpirArrc=pi/4+kpirArr`

`c_1=pi/4`

and

`c_2=pi/4+pi=5/4pi`.

(both the values are `in[0,2pi]`).