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 ![]() ![]() |
Fact (1) | ![]() ![]() |
Since ![]() ![]() | |
2 | If the right hand derivative of ![]() ![]() |
Fact (3) | ![]() ![]() |
Since ![]() ![]() | |
3 | If the (two-sided) derivative of ![]() ![]() |
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 ![]() ![]() |
Fact (2) | ![]() ![]() |
Since ![]() ![]() | |
2 | If the right hand derivative of ![]() ![]() |
Fact (4) | ![]() ![]() |
Since ![]() ![]() | |
3 | If the (two-sided) derivative of ![]() ![]() |
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. |