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?

Jun 22, 2015

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

Explanation:

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. $y = 2 - 20 x + 2 {x}^{2}$ is a function that is continue in all $\mathbb{R}$, and so it is in $\left[4 , 6\right]$;
2. $y ' = - 20 + 4 x$ is a function continue in all $\mathbb{R}$, so our function is derivable in all $\mathbb{R}$, so it is in $\left[4 , 6\right]$;
3. $f \left(4\right) = f \left(6\right) = - 46$.

To find $c$, we have to solve:

$y ' \left(c\right) = 0 \Rightarrow - 20 + 4 c = 0 \Rightarrow 4 c = 20 \Rightarrow c = 5$

(the value is $\in \left[4 , 6\right]$).