The second partial derivative test is a generalization of the second derivative test to multivariable functions and can be used to determine if a critical point is a local minima, local maxima or saddle point.

Definition

Consider the multivariable function with a continuous second order partial derivatives in a neighbourhood around a critical point . The nature of the critical point can be determined by evaluating the following quantity:

Let be the Hessian matrix of at the point :

Then, define the quantity as the determinant of the Hessian matrix:

Then because we know that for continuous functions, the mixed partial derivatives are equal (i.e., ), we can simplify to:

The second partial derivative test states that:

1. If and , then has a local minimum at .
2. If and , then has a local maximum at .
3. If , then has a saddle point at .
4. If , the test is inconclusive, and further analysis is required to determine the nature of the critical point.