Question

True or false

If f'(c) = 0 and f9c0 is not a local maximum, then f(c) is a local minimum.

Answer 1

thats supposed to be f(c)

Answer 2

False.

When f'(c)=0, three things can happen:
1. f(c) is a maximum,
2. f(c) is a stationary point, or
3. f(c) is a minimum.

Cases 1-3 can be distinguished by checking if f"(c) is <, = or > 0.

Answer 3

but why is it not a minimum?

Answer 4

When f'(c)=0 and is not a local maximum, it <i>could</i> be a local minimum, <i>or</i> a stationary point.

A stationary point is where f'(c)=0 and f"(c)=0, i.e. where f'(c-) has the same sign as f'(c+).

An example of a stationary point is f(x)=x³ at x=0.

See:
http://img163.imageshack.us/img163/4127/1292376918.png

Answer 5

so is it a stationary point then?

Answer 6

Neither. Since it can be either a minimum or a stationary point, we cannot say that it is a minimum. So the statement that "it is a minimum" is false.

Consider the statement:
"if x is not greater than zero, then x is less than zero"
is false, because x can be equal to zero.