How do you know whether Rolle's Theorem applies for f(x)=x^2/3+1f(x)=x23+1 on the interval [-1,1]?

1 Answer
Apr 18, 2015

Ask your self whether the hypotheses of Rolle's Theorem are true for this function on this interval:

for f(x)=x^2/3+1f(x)=x23+1 on the interval [-1,1]

H1

Is ff continuous on [-1,1][1,1]?

Yes, ff is a polynomial (quadratic) hence, continuous everywhere. So, in particular, ff is continuous on [-1,1][1,1]

H2

Is ff differentiable on (-1,1)(1,1)?

Yes, f'(x) = (2x)/3 exists for all x in (-1,1)
(In fact, f'(x) exists for all real x, but the theorem only requires the interval.)

H3

Is f(-1) = f(1)?

Yes, either by arithimetic (f(-1) = 4/3 = f(1)

or by using the fact that f is an even function, so f(-x) = f(x) for every x.

Because it satisfies all of the hypotheses, we say that the Theorem applies to this function on this interval.