Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

f(x) = x^3 − x^2 − 2x + 5, [0, 2]

Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

f(x) = x^3 − x^2 − 2x + 5,    [0, 2]

anonymous · Answer

Hi.  What do you know about Rolle's Theorem?

anonymous · Answer

f is a polynomial and it is continuous

anonymous · Answer

so first we find the derivative, 3c-2c-2 ?

anonymous · Answer

actually it would be 
$$f'(x)=3x^2-2x-2$$

anonymous · Answer

It looks like the 3 hypotheses are that the function is continuous over the interval, differential over the interval, and has the same value at the two endpoints

anonymous · Answer

the last one is easiest so let's start there - how do you show that property?

anonymous · Answer

i mean, how do you show it's true for this particular function over this particular interval?

anonymous · Answer

the quadratic formula

anonymous · Answer

what ktklown said. so now you need to compute 
$$f(0),f(5)$$ and see if they are equal.  if so, set 
$$f'(x)=3x^2-2x-2=0$$ and solve for x, you should find one in answer in the interval (0,5)

anonymous · Answer

do i use the quadratic formula now?

anonymous · Answer

yes unless by some miracle it factors

anonymous · Answer

btw i made two mistakes, it should be 
$$f(0)=f(2)$$ and also your answer should be in the interval 
$$(0,2)$$

anonymous · Answer

is a just 3 here?

anonymous · Answer

hello