Question

How do you determine if rolles theorem can be applied to `f(x) = 2 − 20x + 2x^2` on the interval [4,6] and if so how do you find all the values of c in the interval for which f'(c)=0?

Answer 1

It is possible to apply and the answer is `c=5`.

Explanation:

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=2-20x+2x^2` is a function that is continue in all `RR`, and so it is in `[4,6]`;
2. `y'=-20+4x` is a function continue in all `RR`, so our function is derivable in all `RR`, so it is in `[4,6]`;
3. `f(4)=f(6)=-46`.

To find `c`, we have to solve:

`y'(c)=0rArr-20+4c=0rArr4c=20rArrc=5`

(the value is `in[4,6]`).