Point of local extremum implies critical point
Statement
Suppose is a function of one variable and is a point in the interior of the domain of (i.e., is defined on an open interval containing ).
Suppose further that is a point of local extremum for , i.e., attains a local extreme value (either a local maximum or a local minimum) at .
Then, is a critical point for , i.e., either the derivative equals zero or the derivative does not exist.
Facts used
- Local minimum from the left implies left hand derivative is nonpositive if it exists
- Local maximum from the left implies left hand derivative is nonnegative if it exists (has full proof + video)
- Local minimum from the right implies right hand derivative is nonnegative if it exists
- Local maximum from the right implies right hand derivative is nonpositive if it exists
The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (2).
Proof
Local minimum case
Given: A function , a point in the interior of the domain of such that attains a local minimum at , i.e., for all for some choice of .
To prove: If exists, then
Proof
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | If the left hand derivative of at exists, then it is nonpositive, i.e., it is negative or zero. | Fact (1) | attains a local maximum at | Since attains a two-sided local minimum at , it in particular attains a local minimum from the left. Thus, Fact (1) applies. | |
2 | If the right hand derivative of at exists, then it is nonnegative, i.e., it is positive or zero. | Fact (3) | attains a local maximum at | Since attains a two-sided local minimum at , it in particular attains a local minimum from the right. Thus, Fact (3) applies. | |
3 | If the (two-sided) derivative of exists at , it must be zero. | Steps (1), (2) | For the (two-sided) derivative to exist, both the left hand derivative and the right hand derivative must exist and they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero. |
Local maximum case
Given: A function , a point in the interior of the domain of such that attains a local maximum at , i.e., for all for some choice of .
To prove: If exists, then
Proof
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | If the left hand derivative of at exists, then it is nonnegative, i.e., it is positive or zero. | Fact (2) | attains a local maximum at | Since attains a two-sided local maximum at , it in particular attains a local maximum from the left. Thus, Fact (2) applies. | |
2 | If the right hand derivative of at exists, then it is nonpositive, i.e., it is negative or zero. | Fact (4) | attains a local maximum at | Since attains a two-sided local maximum at , it in particular attains a local maximum from the right. Thus, Fact (4) applies. | |
3 | If the (two-sided) derivative of exists at , it must be zero. | Steps (1), (2) | For the (two-sided) derivative to exist, both the left hand derivative and the right hand derivative must exist and they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero. |