Constraint Optimisation

f(x,y) = 5 - (x-2)^2 -2(y-1)^2; Constrain: x + 4y -3 = 0

g(x,y) = f(x,y) + K*(x + 4y -3) where K is the lagrangian Multiplier

dg/dx = -2x + 4 + K = 0 (1)

dg/dy = -4y+4+ 4K = 0 (2)

dg/dK = x + 4y -3 = 0 (3)

Solving (1) & (2), we get

y-2x + 3 = 0 (4)

Solving (3) & (4), we get,

y = -1; x = 7.

Thus for the above values of x and y, we get the optimised value for the above constrained problem