Question

Which is true?
a) If f is continuous at x=c, then f ' (c) exists
b)if f ' (c)=0, then f has a local max or a local min at x=c
c)if f " (c)=0, then f has an inflection point at (c, f(c))
d)If f is differentiable at x=c, then f is continuous at x=c
e)If f is continuous on (a,b), then f has a max on (a,b)

I'm never sure what is right and what is wrong on these. I need help finding the answer and explain how it is that answer.

Answer 1

Your best bet is to think of a counterexample for each:

For a. it might be:

Answer 2

(a) is false because of cusps. The absolute value function is not differentiable at x=0 even though it is continuous there. Check out the graph of y = |x|.
For (b), I think it's true but I'm not sure how to prove it. Maybe Rolle's Theorem or Mean Value Theorem explains it.
(c) is false. Look up the 2nd derivative test. The test is inconclusive if f"(c) = 0. Three functions explain this. y = x^4 ; y= -x^4 and y = x^3 .The 2nd deriv of all three of these is 0 at x = 0. Yet the first one has a relative minimum there, the 2nd one has a relative maximum there and the third has an inflection point.
(d) is true. If it's differentiable then it's continuous for sure. Look at the limit definition of a derivative. The limit exists if the limit approaching from the right exists AND is equal to the limit approaching from the left (which also must exist). So if f is differentiable at x = c, then it must be continuous.
I think (e) is true because of Rolle's Theorem.