Rolle's Theorem

[Mightyducks85]

I need some help answering this for a test I took. We get points back on our exams if we correct our mistakes. Thanks guys.

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.

 

[Deleted member 4993]

I need some help answering this for a test I took. We get points back on our exams if we correct our mistakes. Thanks guys.

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.

Please tell us:

What is the hypothesis of Rolle's theorem for a given function within a domain.

 

[Mightyducks85]

Since f(a) is not equal to f(b), Rolle's Theorem can't be satisfied?

 

[Deleted member 4993]

Since f(a) is not equal to f(b),

Are you sure ?? How did you get that?

Rolle's Theorem can't be satisfied?

 

[Mightyducks85]

f(-1)= (-1^4)-2(-1^2)= 1

f(1)= (1^4)-2(1^2)= -1

So, f(a) doesn't equal f(b)

 

[BigGlenntheHeavy]

\(\displaystyle Let \ f \ be \ continuous \ on \ the \ closed \ interval \ [-1,1] \ and \ differentiable \ on \ the \ open \ interval \ (-1,1).\)\(\displaystyle If \ f(-1) = f(1) \ then \ there \ is \ at \ least \ one \ number \ c \ in \ (-1,1) \ such \ that \ f' \ (c) = 0.\)

\(\displaystyle f(-1) = f(1) = -1, \ ergo \ f' \ (x) = 4x^{3}-4x = 4x(x^{2}-1) = 0. Hence \ x = 0,-1,1. \ Discard \ 1,-1,\)

\(\displaystyle Therefore, \ f' \ (0) = 0, \ c = 0.\)

Factoid: Rolle's Theorem - This theorem that bears his name appeared in a little known treatise on geometry and algebra entitled Methode pour resoudre les egalites published in 1691. It is ironic that one of the basic results in the theory of calculus was proved by a person who was vigorously opposed to the calculus methods of his contemporaries. In later life, Rolle acknowledged that the calculus techniques were of value and basically sound.