Question

Use the method of Lagrange multipliers to optimize the function subject to the given constraint.
f(x, y) = 4x + 6y - 2x2 - 7y2 where 3x + 9y = 9
Find the coordinates of the optimal point subject to the constraint.

Answer 1

ok , where are you getting stuck at with this?

Answer 2

i need the value for x and the value for y and f(x,y) but I dont know where to start...

Answer 3

well, we can start by creating a single function from these F(x,y,L). Either by adding or subtracting the consttraint

Answer 4

like this:

F(x, y,L) = 4x + 6y - 2x2 - 7y2 + L(3x + 9y - 9)

Answer 5

that is the start :)

we can distribute the L thru the () .... i used L to represent the Lagrange multiplier

Answer 6

okk thanks is L lambda?

Answer 7

yes, L = lambda :)

F(x,y,L) = 4x + 6y - 2x2 - 7y2 + L3x + L9y - L9

Now we find our partials:

Fx
Fy
FL

Answer 8

how would you use the close point method to classify the optimal point.
to find out if the optimal point is a max or min?

Answer 9

close point sounds like you find point near it and determine whether they rise of fall

Answer 10

i dont think ive ever actually tried the "method" but thats what it sounds like

Answer 11

oh ok sounds good thanks for your help ! im gona try to see if i can figure it out now :/

Answer 12

Fx = 4x -2x2 + L3x
= 4 -4x + L3 = 0; solve for x

Fy = 6y -7y2 +L9y
= 6 - 14y +L9 = 0; solve for y

FL = L3x+L9y-L9
= 3x +9y -9 = 0; substitute in x and y and solve for L

Answer 13

ok thank you so so much!

Answer 14

let me know what you get for an answer if you want :) and i can dbl chk it