For a univariate function , a second derivative test tells:
- if is concave up (if )
- if is concave down (if )
- where inflection points are (where concavity changes)
If is a multivariate function, to determine whether has a minimum, maximum, or saddle point at the critical point , we can use the second derivative test.
First find , , and (, , ). Then calculate the determinant as below:
Then:
- If and , then has a local minimum at .
- If and , then has a local maximum at .
- If , then has saddle point at .
- If then test is inconclusive.