Question

Which one of the following statements is true?

If f ′′(x) > 0 on the interval (a, b) then f(x) is concave down on the interval (a, b).
If f ′(x) > 0 on the interval (a, b) then f(x) is increasing on the interval (a, b).
If f ′(c) = 0, then x = c is a relative maximum on the graph of f(x).
None of these are true.

Answer 1

the first derivitive is the slope
a positive 1st deriviive is positive slope or increasing
a negaitve 1st derivitive is negative slope or decreasing

the 2nd derivitive tells about the concavity
a negative second derivitive means it is concave down at that point
a positive 2nd derivitive means it is concave up at that point

so

first one is false
2nd is true
3rd is false because it could possibly be a minimum as well

answer is 2nd one