MVT/Rolle's Theorem: Defining Intervals

[pyrosilver]

Homework Statement

Define interval where a) MVT applies, b) Rolle's applies, c) MVT doesn't apply. explain. (In this, we're saying Rolle's is f(a) = f(b), not f(a) = (f(b) = 0.

Homework Equations

The Attempt at a Solution

f(x) = (x-4)^(2/3)

a) for MVT, must be cont on [a,b] and diff on (a,b). It is not diff at x=4, but for interval [5,8] it should be fine, so a = [5,8]?
b) I'm confused for this one. f(1) = 2.08. f(7) = 2.08. However, x is not diff at 4, so [1,7] isn't a valid interval, right? What is a valid interval, one that let's Rolle's apply?
c) MVT doesn't apply on [1,7] because x is not diff at 4? Is that okay?

Please help!

 

[HallsofIvy]

Homework Statement

Define interval where a) MVT applies, b) Rolle's applies, c) MVT doesn't apply. explain. (In this, we're saying Rolle's is f(a) = f(b), not f(a) = (f(b) = 0.

Homework Equations

The Attempt at a Solution

f(x) = (x-4)^(2/3)

a) for MVT, must be cont on [a,b] and diff on (a,b). It is not diff at x=4, but for interval [5,8] it should be fine, so a = [5,8]?

Yes, any interval that does not include 4 works.

b) I'm confused for this one. f(1) = 2.08. f(7) = 2.08. However, x is not diff at 4, so [1,7] isn't a valid interval, right? What is a valid interval, one that let's Rolle's apply?

You are correct that [1, 7] does not work. You want to find some a, b, such that a and b are both less than 4 or both larger than 4 such that f(a)= f(b).
It is easy to see that f(x) is increasing for x> 4 and decreasing for x< 4. That is, in order that f(a)= f(b), either a= b or one is less than 4 and the other larger than 4. Neither gives a valid interval. Notice that if there were a valid interval, Rolle's theorem would say that f'(c)= 0 at some point inside that interval. And f' is NEVER 0 for this function. There is no such interval.

c) MVT doesn't apply on [1,7] because x is not diff at 4? Is that okay?

Yes, or any interval including 4.

Please help!