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 , we

  1. find the critical points, i.e. the solutions of
  2. 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 and we can remove other terms in the Taylor expension:

  • 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 is a local maximum, minimum, or saddle point, we calculate the Hessian:

(Note that if we assume the derivatives are continuous)

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

See Also

  • Taking second derivative is also useful to decide whether a real function is convex or concave at a point