# 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?

May 29, 2015

The Rolles theorem says that if:

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

then at least one $c \in \left(a , b\right)$ as if $f ' \left(c\right) = 0$ exists.

So:

1. $y = 2 {x}^{2} - 5 x + 1$ is a function that is continue in all $\mathbb{R}$, and so it is in $\left[0 , 2\right]$;
2. $y ' = 4 x - 5$ is a function continue in all $\mathbb{R}$, so our function is derivable in all $\mathbb{R}$, so it is in $\left[0 , 2\right]$;
3. f(0)=1;f(2)=-1rArrf(a)!=f(b) and so we can't apply the Rolles Theorem.