Question

Determine whether the following statements are true or false.
a) If f'(x)=0 at each x of an open interval (a, b), then f is constant on (a, b).
b) Suppose that f is twice differentiable on an open interval I. If f' increases on I, then the graph of is concave up.
c) Suppose that f is twice differentiable on an open interval I. If f' decreasing on I, then the graph of is concave down.
d) For c in I, if f'(c)=0 and f"(c)<0 then f has a local minimum at x=c.
e) If the point (c, f(c)) is a point of inflection, then f"(c)=0.

Answer 1

If f' increases over an interval it means f'' will be positive over that interval.
If f''(x) is positive over an interval it means f(x) will be concave up over the interval.

Answer 2

a) is true.

Answer 3

It it because it's neither increasing nor decreasing?

Answer 4

f'(x0 = 0 over an interval means the slope of f(x) is 0 over that interval.
That means it is a line parallel to the x axis and is a constant.

Answer 5

I get that one now, could you help me with the others too please?

Answer 6

The first three are true.
Fourth is false. For c in I, if f'(c)=0 and f"(c)<0 then f has a local minimum at x=c. No. It will be a local maximum
Fifth is true. (But the converse is not always true).

Answer 7

You said before that if f"(x) is positive over an interval then it's concave up so the converse should be true for that right?

Answer 8

Yes.

Answer 9

And for the fourth one, how do you know that it's a local maximum and not a minimum?

Answer 10

And the fifth is true because you find the point of inflection by solving the 2nd derivative for 0 right?

Answer 11

A local minimum occurs in a concave up curve. f''(c) > 0
A local maximum occurs in a concave down curve. f''(c) < 0

Answer 12

f''(c) = 0 is a necessary condition for a point of inflection at x = c.
It is NOT a sufficient condition. That is f''(c) = 0 does not necessarily mean there is a point of inflection at x = c.

Answer 13

I understand now, thank you

Answer 14

you are welcome.