Question

Use Lagrange multipliers to solve the given optimization problem.

Find the maximum value of
f(x, y) = xy
subject to x + 2y = 52.
@whpalmer4

Answer 1

Sure, now I have to learn about Lagrange multipliers, eh? :-)

Answer 2

haha i just keep testing you!

Answer 3

calc 3?

Answer 4

idk if its labeled as calc 3 but calculus yeah!

Answer 5

ahh ok

Answer 6

say, g is the constraint function,
then the min/max value occurs when :
\(\bigtriangledown f = \lambda \bigtriangledown g\)

find the partials fx, fy, gx, gy and setup below equations :
\(f_x = \lambda g_x\)
\(f_y = \lambda g_y\)

and solve

Answer 7

u able to find \(\lambda\), x, y values?

Answer 8

wid the constraint equation,
u have 3 equations and 3 unknowns to find.. piece of cake to solve

Answer 9

$$
f(x, y) = xy\\
g(x,y)= x + 2y = 52\\
\Lambda(\lambda,x,y)=f(x,y)-\lambda(g(x,y)-c)\\
\Lambda(\lambda,x,y)=xy-\lambda(x+2y-52)\\
\Lambda_x=y-\lambda=0\\
\Lambda_y=x-2\lambda=0\\
\Lambda_{\lambda}=x+2y-52=0\\
$$
Solve for x and y.

Answer 10

$$
y=\lambda\\
x-2y=0\\
x=2y\\
2y+2y-52=0\\
4y=52\\
y=\cfrac{52}{4}\\
$$
That's it!

Any questions?

Answer 11

no that all makes sense!! thank you :)

Answer 12

yw