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Section 15.7

- Maximum and Minimum Values

MAXIMA AND MINIMA

A function of two variables has a local maximum

at (a, b) if f (x, y) f (a, b) when (x, y)

is near (a, b). The number f (a, b) is called

a local maximum value. If f (x, y) f (a, b)

when (x, y) is near (a, b), then f (a, b) is a

local minimum value.

If the inequalities above hold for all points (x,

y) in the domain of f, then f has an absolute

maximum (or absolute minimum) at (a, b).

Theorem If f has a local maximum or minimum

at (a, b) and the first-order partial derivatives

of f exist there, then fx(a, b) 0 and fy(a,

b) 0

CRITICAL POINTS

A point (a, b) is called a critical point (or

stationary point) of f if fx(a, b) 0 and

fy(a, b) 0, or one of these partial derivatives

does not exist.

SECOND DERIVATIVES TEST

Suppose the second partial derivatives of f are

continuous on a disk with center (a, b), and

suppose that fx(a, b) 0 and fy(a, b) 0.

Let (a) If D gt 0 and fxx(a, b) gt 0, then f

(a, b) is a local minimum. (b) If D gt 0 and

fxx(a, b) lt 0, then f (a, b) is a local

maximum. (c) If D lt 0, then f (a, b) is not a

local maximum or minimum.

NOTES ON THE SECOND DERIVATIVES TEST

- In case (c) the point (a, b) is called a saddle

point of f and the graph crosses its tangent

plane. (Think of the hyperbolic paraboloid.) - If D 0, the test gives no information f

could have a local maximum or local minimum at

(a, b), or (a, b) could be a saddle point. - To remember this formula for D, it is helpful to

write it as a determinant

CLOSED SETS BOUNDED SETS

- A boundary point of a set D is a point (a, b)

such that every disk with center (a, b) contains

points in D and also point not in D. - A closed set in is one that contains all

its boundary points. - A bounded set in is one that is contained

within some disk.

EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO

VARIABLES

If f is continuous on a closed, bounded set D

in , then f attains an absolute maximum

value f (x1, y1) and an absolute minimum value

f (x2, y2) at some points (x1, y1) and (x2, y2)

in D.

FINDING THE ABSOLUTE MINIMUM AND MAXIMUM

To find the absolute maximum and minimum values

of a continuous function f on a closed bounded

set D

- Find the values of f at the critical points of

D. - Find the extreme values of f on the boundary of

D. - The largest of the values from steps 1 and 2 is

the absolute maximum value the smallest of

these values is the absolute minimum value.

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