How do you determine if rolles theorem can be applied to $f (x) = 2 - 20 x + 2 x^{2}$ $f(x) = 2 − 20x + 2x^2$ 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?

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Answer 1 · 2015-06-22T18:30:22

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

The Rolles theorem says that if:

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

So:

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

To find $c$ , we have to solve:

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

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

How do you determine if rolles theorem can be applied to f(x) = 2 − 20x + 2x^2f(x)=2−20x+2x2f(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?