Find all maxima and minima of the function fx y exy subject

Find all maxima and minima of the function f(x, y) = e^xy subject to the constraint x^3 + y^3= 16.

Solution

Given that

f(x , y) = e^xy subject to the constraint x³ + y³ = 16

Let g(x , y) =  x³ + y³ = 16

Using legrange multipliers finding maxima and minima

f(x,y) = g(x,y)

< ye^xy , xe^xy > = < 3x² , 3y² >

  ye^xy = 3x² , xe^xy = 3y²

   =  ye^xy/ 3x^{2 ,} = xe^xy / 3y²

Hence,

   ye^xy/ 3x² = xe^xy / 3y²

On solving ,

x/y =1

x = y

x³ + y³ = 16

Substitute x = y in this equation

x³ + x³ = 16

2x³ = 16

x³ = 16/2

x³ = 8

x = 2

Hence,

x = y

y = 2

Substitute x = 2 , y = 2 in f(x , y) = e^xy

   f(2,2) = e^2.2

      f(2,2) = e⁴

      f(2,2) = 54.598

Therefore,

The maxima of given function = 54.598 at  f(2,2)

The given function has no minimum