Question

find the maximum and minimum value of xy(a-x-y)​

Answer 1

Answer:

A function f(x, y) has a local maximum at (a, b) if f(x, y) ≤ f(a, b) for all (x, y) in

some disk with center (a, b). The number f(a, b) is called a local maximum value.

Similarly, f(x, y) has a local minimum at (a, b) if f(x, y) ≥ f(a, b) for all (x, y) in

some disk with center (a, b). The number f(a, b) is called a local minimum value.

A local maximum or minimum is called a local extremum.

• A function f(x, y) has an absolute maximum at (a, b) if f(x, y) ≤ f(a, b) for all

(x, y) in the domain of f. The number f(a, b) is called an absolute maximum

value. Similarly, f(x, y) has an absolute minimum at (a, b) if f(x, y) ≥ f(a, b) for

all (x, y) in the domain of f. The number f(a, b) is called an absolute minimum

value. An absolute maximum or minimum is called an absolute extremum.

Theorem: (Partial Derivative Criteria)

If f has a local extremum at (a, b) and the first-order partial derivatives of f exist at (a, b),

then fx(a, b) = fy(a, b) = 0.

Definition: A point (a, b) such that fx(a, b) = fy(a, b) = 0, or one of these partial derivatives

does not exist, is called a critical point of f.

Example: Find all critical points of f(x, y) = x

2 + y

2 + 4x − 6y + 1.

The first-order partial derivatives are

fx(x, y) = 2x + 4 and fy(x, y) = 2y − 6.

Setting these partial derivatives equal to zero, we have x = −2 and y = 3. Thus, (−2, 3) is

the only critical point.