Mean Value Theorem/Rolle's Theorem and differentiability

[NanaToru]

Homework Statement

Let f(x) = 1 - x^2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem?

Homework Equations

The Attempt at a Solution

f(x) = 1 - x^2/3.
f(-1) = 1 - 1 = 0
f(1) = 1 - 1 = 0

f' = 2/3 x ^-1/3.

I don't understand why this doesn't have a number c in f'(c), or why Rolle's theorem excludes nondifferentiable points?

 

[LCKurtz]

Homework Statement

Let f(x) = 1 - x^2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem?

Homework Equations

The Attempt at a Solution

f(x) = 1 - x^2/3.
f(-1) = 1 - 1 = 0
f(1) = 1 - 1 = 0

f' = 2/3 x ^-1/3.

I don't understand why this doesn't have a number c in f'(c), or why Rolle's theorem excludes nondifferentiable points?

For a function to violate Rolle's theorem, it would need to do two things:
1. Satisfy all the hypotheses of Rolle's theorem.
2. Fail to satisfy the conclusion of Rolle's theorem.

Does this function do those two things?

 

[NanaToru]

Hm... I'm not sure it satisfies all the hypotheses--from what the back of the book says, it isn't differentiate on the interval of (-1, 1) but I'm not sure how? Did I do a bad job differentiating it?

 

[LCKurtz]

What do you get for f'(0)?

 

[NanaToru]

Is it not 0? Or is that not a valid answer?

 

[LCKurtz]

Is it not 0? Or is that not a valid answer?

If you think $\frac 1 0 = 0$ then you must also think $0\cdot 0 = 1$?

 

[NanaToru]

...This is honestly the most embarrassing moment of my life. Thank you though!

 

[HallsofIvy]

...This is honestly the most embarrassing moment of my life. Thank you though!

Just wait!