Do local extrema occur if and only if f'(x) = 0?
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
Counterexample 1:
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.
Consider the example of , which is continuous and differentiable everywhere for all
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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
Counterexample 2:
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
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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.
True or False?
The local extrema of are the points where .
The graph at right depicts the function in the interval .
How many local extrema does the function have if its domain is restricted to
See Also