Optimization problem Analytically find the optimum solutions

Optimization problem.

Analytically, find the optimum solutions and optimum function variable to the following optimization problem. min_x, y f(x, y) = (x - 2)^2 + (y + 1)^2 subject to: 2x + 3y - 4 = 0

Solution

We solve this problem using method of Lagrangians

Let, g(x,y)=2x+3y-4=0

Sol Lagrangian for htis problem is

L=f(x,y)+t g(x,y)

L_x=2(x-2)+2t=0

L_y=2(y+1)+3t=0

HEnce,

3(x-2)=4(y+1)

(x-2)=4(y+1)/3

x=2+4y/3+1/3=4y/3+7/3

x=4y/3+7/3

2x+3y=4

8y/3+14/3+3y=4

8y+14+9y=12

17y=-2

y=-2/17

x=4y/3+7/3

x=-8/51+7/3=(111)/51=37/17

f(x)_min=(37/17-2)^2+(-2/17+1)^2=(3/17)^2+(15/17)^2=9/289+225/289=234/289