1. ## [SOLVED] Rolle's Theorem

Show that f(x)=x^4-2x^2 satisfies the hypothesis of Rolle's Theorem on [-1,1]. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

I believe Rolle's Thm states that f(a) has to equal f(b) to find a number c... I calculated f(a) to be 1 and f(b) to be -1, therefore I can't find c, right? Or did I do something wrong?

2. Originally Posted by Mightyducks85
I believe Rolle's Thm states that....
Gracious! They were supposed to give you the Theorem in your book!

Rolle's Theorem states that, if a function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists some x = c within the interval (a, b) such that f'(c) = 0.

Originally Posted by Mightyducks85
Show that f(x)=x^4-2x^2 satisfies the hypothesis of Rolle's Theorem on [-1,1]. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
To "show" that the given function "satisfies the hypothesis" (the "is continuous", "is differentiable", and "is equal-valued" bits), you need to show or state that each of these "givens" or if-statements is true. Are polynomials continuous? (You should have a rule or assumption for this.) Are polynomials differentiable? (You should have a rule or assumption for this.) Does f(-1) equal f(1)? (For this, evaluate f(x) at x = -1 and at x = 1, and compare their values.)

Then, once that's done, you can apply the "then" part. What does Rolle's Theorem say that the value of f(c) will be, for some x = c?

3. Originally Posted by Mightyducks85
Show that f(x)=x^4-2x^2 satisfies the hypothesis of Rolle's Theorem on [-1,1]. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

I believe Rolle's Thm states that f(a) has to equal f(b) to find a number c... I calculated f(a) to be 1 and f(b) to be -1, therefore I can't find c, right? Or did I do something wrong?
Yes, you did something wrong. $f(-1)= (-1)^4- 2(-1)^2= 1- 2= -1$, not 1.