Question

How do you determine if rolles theorem can be applied to `f(x) = 2x^2 − 5x + 1` on the interval [0,2] 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=2x^2-5x+1` is a function that is continue in all `RR`, and so it is in `[0,2]`;
2. `y'=4x-5` is a function continue in all `RR`, so our function is derivable in all `RR`, so it is in `[0,2]`;
3. `f(0)=1;f(2)=-1rArrf(a)!=f(b)` and so we can't apply the Rolles Theorem.