True or false
If f'(c) = 0 and f9c0 is not a local maximum, then f(c) is a local minimum.
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.
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
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.
False.
If f'(c) = 0 and f''(c) > 0, then f(c) is a local minimum. However, if f''(c) < 0 or f''(c) does not exist, we cannot determine whether f(c) is a local minimum or maximum based solely on the information provided.
To understand why, let's explain the concepts involved:
1. First derivative test: The first derivative of a function f(x) gives us information about the slope of the function at any given point. If f'(c) = 0, it means that the function is either experiencing a change from increasing to decreasing or vice versa at point c. However, we cannot determine the exact nature of the point without considering the second derivative.
2. Second derivative test: The second derivative of a function f(x) gives us information about the concavity of the function at any given point. If f''(c) > 0, it means that the function is concave up or has a local minimum at point c. If f''(c) < 0, it means that the function is concave down or has a local maximum at point c. However, if f''(c) = 0 or f''(c) does not exist, the test is inconclusive.
In our case, we know that f'(c) = 0, which suggests a potential turning point at c. However, without knowing the concavity, as indicated by f''(c), we cannot determine whether f(c) is a local minimum or maximum. Therefore, the statement is false.