Lagrange Multipliers

Question

Use the method of Lagrange multipliers to maximize and minimize the function subject to the given constraints. Maximize and minimize the function f(x, y) = e^xy subject to the constraints x² + y² = 8 and x ≥ 0.



Answer 1

Hi,

 f(x, y) = e^xy subject to the constraints x² + y² = 8 and x ≥0

As you know:∇

∇f(x,y)=λ∇g(x,y) ======> g(x,y) = x² + y² -8=0 , C= 0

∇f(x,y) = ( f'_x, f'_y) = (ye^xy, xe^xy) AND λ∇g(x,y) = λ(g'_x, g'_y) = λ( 2x,2y)

(ye^xy, xe^xy) = λ( 2x,2y) =====> ye^xy= 2λx I AND xe^xy= 2λy II

make a system with I and II and solve it by divide two equations

y/x = x/y ====> x²=y² Then replace to the g(x,y) and solve it

x² + y² = 8 ====> y²  + y² = 8 ===> y = ±2 -----> x = ±2 base of given information

x = 2 acceptable and y = ±2 then replace to the function

f(x,y) = e^xy

the answers will be f(2,2) = e ⁴ and f(2,-2) = e ^-4

I hope it is useful,

Minoo