True or false
If f'(c) = 0 and f9c0 is not a local maximum, then f(c) is a local minimum.
6 answers
thats supposed to be f(c)
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.
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.
but why is it not a minimum?
When f'(c)=0 and is not a local maximum, it could be a local minimum, or 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
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
so is it a stationary point then?
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.
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.
1 answer