Rolle's theorem

Question

Determine whether Rolle's theorem applies to the function shown below on the given interval. if so, find the point that are guaranteed to exists by Rolle's theorem.
f(x)=1-x^2/3 ;[-1,1]

Answer 1

Rolle's Theorem states that if a function is continuous and differentiable over an interval [a,b] and f(a) = f(b) then somewhere in the interval there must be a "flat" point at x=c, where f'(c) = 0.

 

f(x) = 1 - x²/3

This is a polynomial, so it is continuous and differentiable everywhere.

f(-1) = 1 - (-1)²/3 = 1 - 1/3 = 2/3

f(1) = 1 - 1²/3 = 1 - 1/3 = 2/3

This function satisfies the conditions of Rolle's Theorem.

 

f'(x) = -2/3 x

f'(c) = -2/3 c = 0

c = 0

 

c = 0 ∈ [-1,-1] satisfies Rolle's Theorem.