Sahithyan's S2 — Methods of Mathematics
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 .
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
A critical point in which it is not a local maximum or local minimum.
Closed interval method
A method for finding absolute minimum and absolute maximum values of which is continuous on a closed, bounded set .
- Find the values of at the critical points of in the interior of .
- Find the extreme values of on the boundary of .
- Maximum and minimum values from step 1 and 2 are the absolute maximum and absolute minimum.