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:

  1. If and , then has a local minimum at .
  2. If and , then has a local maximum at .
  3. If , then has saddle point at .
  4. If then test is inconclusive.