Local maximum from the left implies left hand derivative is nonnegative if it exists
This article describes a test that can be used to determine whether a point in the domain of a function gives a point of local, endpoint, or absolute (global) maximum or minimum of the function, and/or to narrow down the possibilities for points where such maxima or minima occur.
View a complete list of such tests
Further, suppose that the left hand derivative of at exists. Then, this left hand derivative is nonnegative, i.e., it is either positive or zero.
- Local maximum from the right implies right hand derivative is nonpositive if it exists
- Local minimum from the left implies left hand derivative is nonpositive if it exists
- Local minimum from the right implies right hand derivative is nonnegative if it exists
To prove: If , i.e., the left hand derivative of at exists, then it is positive or zero, i.e., .
Proof: The left hand derivative is the left hand limit of the difference quotient, i.e., it is given as:
We proceed as follows:
|Step no.||Assertion||Given data used||Previous steps used||Explanation|
|1||To take the limit as , it suffices to restrict attention to , i.e., we only care about the behavior of on the immediate left of||Follows from the definition of limit|
|2||For , the denominator is negative, and the numerator is non-positive, i.e., it is either negative or zero||for .||The denominator is negative because . The numerator is positive or zero because for .|
|3||For , the difference quotient is non-negative, i.e., it is either positive or zero.||Step (2)||Quotient of a number that is by a number that is must be .|
|4||If exists, it must be non-negative||Step (3)||By Step (3), is the limit of an expression that's . Hence, it must be .|
Note on strictness
Note that even if we assume that is a point of strict local maximum from the left (i.e., that for , we can only conclude that the left hand derivative at , it exists, is positive or zero. We cannot eliminate the possibility of its being zero.
The reason is as follows: we do know that for a strict local maximum from the left, the difference quotient with any point on the immediate left is positive. However, the limit of this quantity can still be zero, because the limit of a function taking positive values can be zero.
The fact that the zero case survives even in the strict situation is crucial to the observation that point of local extremum implies critical point.