Second derivative test can be used to decide whether a critical point is local minimum, maximum, or saddle point.
Second Derivative Test for Functions of One Variable
To find local maximum and minimum of a function
- find the critical points, i.e. the solutions of
- test the second derivative for each critical point
means that is a local minimum means that is a local maximum - If
, the test failed
Second Derivative Test for Several Variables
At a critical point, we know that the first derivative is zero, so the second derivative dominates local behavior for all
- If
for all , local minimum - If
for all , local maximum
We can use eigenvalues to figure out the above condition.
Second Derivative Test for Functions of Two Variables
For a function of just two variables, we can use the “trace-determinant method” to perform second derivative test for function
First we find the critical points by solving the linear equations
Then to see if a critical point
(Note that
Then
- If
, we have a saddle point - If
and the trace , we have a local minimum - If
and , we have a local maximum - Otherwise, we have a degenerated critical point and the test fail
Example: find the critical points of
and determine their type Solution:
To find the critical points we need to solve
And we get
and . At , we have , so it is a saddle point At , we have and , so it is a minimum point.
See Also
- Taking second derivative is also useful to decide whether a real function is convex or concave at a point