local minimum

A local minimum is a critical point within an interval, at which the function has a minimum value.

In a univariate unction

A single-variable function has a local minimum at if:

• and
• OR changes its sign from to through . This ensure that the zero first derivative occurs at the bottom of a bowl

To test whether it’s also an absolute minimum, compare with the function value at endpoints

In a multivariate Function

For a multivariate function , we decide whether is a local minimum using the second partial derivative test

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.

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