Extremums

Revise Extremums from S1.

Theorems

Local extremum

Suppose has a local maximum or minimum at and the first order partial derivatives of exist there. Then and .

Proof Hint

Fix 1 variable and apply the first derivative test.

Extreme Value Theorem

If is continuous on a closed, bounded set then has an absolute maximum and absolute minimum value in .

Implicit Function Theorem

Suppose , and . Then there exists a unique function defined on a neighbourhood of with:

Second Derivatives Test

Suppose the second partial derivatives of are continuous on a disk with center , and suppose that and . Let:

• If , then no information.
• If , then is a local minimum.
• If , then is a local maximum.
• If then is not a local maximum or minimum.

Saddle point

If , then is not a local maximum or minimum. And is a saddle point. The graph of has a crosses its tangent plane at .

Closed interval method

A method for finding absolute minimum and absolute maximum values of which is continuous on a closed, bounded set .

1. Find the values of at the critical points of in the interior of .
2. Find the extreme values of on the boundary of .
3. Maximum and minimum values from step 1 and 2 are the absolute maximum and absolute minimum.