Second Derivative Test

Suppose is a function of that is twice differentiable at a stationary point .

1. If , then has a local minimum at .

2. If , then has a local maximum at .

The extremum test gives slightly more general conditions under which a function with is a maximum or minimum.

If is a two-dimensional function that has a local extremum at a point and has continuous partial derivatives at this point, then and . The second partial derivatives test classifies the point as a local maximum or local minimum.

Define the second derivative test discriminant as

			(1)
			(2)

Then

1. If and , the point is a local minimum.

2. If and , the point is a local maximum.

3. If , the point is a saddle point.

4. If , higher order tests must be used.