Constraint Minimization

Problem statement:
Minimize the function 5-(x-2)²-2(y-1⁾² subject to x+4y−3=0

Solution:

L=f−λg, where f= 5-(x-2)²-2(y-1)² and g=x+4y-3

Thus, I solve

L= 5-(x-2)²-2(y-1)²-λ(x+4y-3)

Lx=-2(x-2)-λ=0 →→ EqnI

Ly=−4(y-1)-4λ=0 →→EqnII

Lλ =x+4y-3=0 →→ EqnIII

which gives x=−λ/2 +2 from Eqn I & y=−λ+1 from Eqn II

Substituting x and y equations in EqnIII:

−λ/2+2+4(1−λ)=3

Thus, λ=2/3 and so x=5/3, y=1/3 ensures f has a minimum subject to g=0.