Minimize f (x,y) = x^2 + y^2 subject to the constraint function g (x,y) = x^2y-16 = 0.

Answer 1

The best approach seems to be the method of Lagrange multipliers.

This works by considering the function   f(x,y) + L g(x,y)  where L is a Lagrange multiplier

Three equations are obtained by setting the partial derivative wrt x equal to zero , setting the partial derivative wrt y equal to zero , with the third equation being g (x,y) = 0.

Explicitly these three are:

2 x + 2x y L = 0

2 y+  x^2  y L = 0

x^2 y = 16

This is a set of three equations with three unknowns ( x, y, L)

The most straightforward method of solution is to use the third equation to get y = 16/x^2 and then substituting into the other two. This leads to a set of two equations with two unknowns (x , L) which is easily solved to get

x = 2^(5/4)    y = 2^(3/2)   L = - 2^(-3/2)

Alternatively, x can be – 2^(5/4)   but that leads to the same minimum value of x^2 + y^2 which is

2^(5/2) + 2^3