Question

Use Lagrange multipliers to solve the given optimization problem.

Find the maximum value of f(x, y)-xy subject to x + 2y 52.
fmax=______
Also find the corresponding point
(x, y)=__________.

Answer 1

fmax = xy = 26 × 13 = 338

(x,y) = (26,13)

Explanation:

Given that:

f(x, y) = xy

subject to x + 2y = 52

So;

x = 52 - 2y

and;

f(x, y) = xy

f(x, y) = (52- 2y) y

f(x, y) = 52y - 2y²

In order to maximize this function;

52y - 2y² = 0

26 y - y² = 0

26 - 2y = 0

-2y = -26

y = -26/-2

y = 13

Again:

x = 52 - 2y

x = 52 - 2(13)

x = 52 - 26

x = 26

fmax = xy = 26 × 13 = 338

(x,y) = (26,13)