This is part of a series on common misconceptions.
True or False?
Local extrema of occur if and only if
Why some people say it's true: That is the first derivative test we were taught in high school.
Why some people say it's false: There are cases that are exceptions to this statement. Also, I was taught that this is just the first step of the first derivative test.
The statement is .
Knowing that a function's derivative is 0 at a specific point does not necessarily mean that there's a local minimum or maximum at that point. When it only implies that there is a horizontal tangent at that point. An example of the case when a function has a horizontal tangent at a point that is not a local extremum is given below.
Additionally, it's possible for a function to have a local minimum or maximum at a point where its derivative is non-zero in a variety of different ways outlined further below. For example, has a local minimum at despite
This is a simple counterexample of the given statement: an example of when the derivative of a function is 0 at a point which is not a local extremum.
Differentiating with respect to we have Setting it to zero, we get But graphically we can clearly see that the function does not attain a local extremum at The curve merely goes from being concave downwards for to being concave upwards for
for is an example of a function which has a local extremum at a point where its derivative is not 0. This occurs because the function has a restricted domain.
Shown below is the graph of for
The function attains its maximum value at and minimum at
However, the derivative of the function is not equal to zero at any of the points:
implying the given statement is false.
More information on local and global extrema can be found here: Extrema.
The following are two common objections to the counterexample given above:
The local extrema of for are
Rebuttal: The gradient of the function is positive at , and therefore it is increasing and the value we got isn't the local maximum.
Reply: The function isn't defined for , and hence we have indeed reached the local maximum at
Rebuttal: Isn't the minimum value of
Reply: The minimum value of the function in the domain is not The value corresponds to the minimum value on a larger domain.