second partial derivative test

second partial derivative test test to determine whether a critical point of a twice-differentiable multivariate function is a local minimum or a local maximum

Suppose all second partial derivatives are defined and continuous on a neighborhood around the critical point of a multivariate function

Define:

• If and , is a local minimum
• If and , is a local maximum
• If , is a saddle point
• If , the test is inconclusive

Examples

Example 1: multivariable optimization

Example 2: The scalar field has a critical point at . How does the second partial derivative test classify this point?

1. Calculate regular partial derivative

2. Calculate all second-order partial derivatives

3. Perform the second partial derivative test:

The test is inconclusive.