Question

Which of the following statements is/are true?
I.If f '(x) exists and is nonzero for all x, then f(1)
II.If f is differentiable for all x and f (-1) = f(1), then there is a number c, such that |c| < 1 and f '(c) = 0.
III.If f '(c) = 0, then f has a local maximum or minimum at x = c.


a.)I only
b.)II only
c.)I and III only
d.)I and II only

Answer 1

Statement I needs rephrasing, not sure what is meant.

Statement II is actually a specific case of the mean value theorem. Remember what the mean value theorem is?

Statement III well think about it, are there pts where you derivative is zero which AREN'T maximum or minimum turning points?

Answer 2

as a bit of extra trivia, the theorem described in 2 is specifically known as Rolle's Theorem, which as mentioned is a specific case of the Mean Value Theorem (and can be used to prove the MVT)

https://en.wikipedia.org/wiki/Rolle's_theorem

Answer 3

yes so III is not true

Answer 4

and statement I got cut off on the assignment so i am not really sure what is mean by that

Answer 5

but then II would be true

Answer 6

II would be true by Rolle's theorem (and extreme value theorem in part).

Since the gradient between the two pts is zero and the function is continuous and differentiable in the interval, you will expect at least one pt where the gradient of the tangent would match the secant of 0.

III yes, horizontal inflexions are not maxima or minima but f'(x) = 0