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Calculus 3

(Week 5)-Max-Min

Note that, if there is a greatest local maximum point, it is calledAbsolute Maximum Point.

On the other hand, if there is a smallest local minimum point, it iscalled Absolute Minimum Point.

Definition.(Critical Point) A point (a, b) is called a critical point of f , if

fx(a, b) = 0 and fy(a, b) = 0

OR

If one of these partial derivatives does not exist.

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Example. Let’s look at local max-min of the following function,

f(x, y) = x2 + y2 − 2x− 6y + 14

Remark. Sometimes, local max, local min values called ’Extreme Values’.

Example. Find the extreme values of the function f(x, y) = y2 − x2.

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Exercises.

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PART 2

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Example. Find the absolute max. and min. values of f on the set D, if

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Exercises.

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Exercises of the Week

1-) Find the local max-min and saddle points (if there is any) of the following.

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2-) Find the absolute max-min points of the following.