Constraint Minimization - Pen and Paper calculation

Minimize the following function

5-(x-2)^2 -2(y-1)^2

subject to the following constraint

x + 4y = 3

u(x,y)=5-(x-2)^2-2(y-1)^2

const=x+4y-3

f(x,y)=u(x,y)+ʎ(const)

f(x,y)=5-(x-2)^2-2(y-1)^2+ʎ(x+4y-3)

df/dx=-2(x-2)1)+ ʎ

df/dx=-2x+4+ʎ   

df/dy=-2*2(y-1)1+4ʎ)

df/dʎ=x+4y-3

df/dʎ=0    df/dy=0     df/dx=0

-2x+4+ʎ=0 --- 1

-4y+4+4ʎ=0 --- 2

x+4y-3=0 - -- 3

Solving eqns 1 – 2

-2x+4y-3ʎ=0 -- 4 

df/dy =-4y+4+4ʎ

Solving eqns 3 – 4

-3x-3-3ʎ=0

x=1-ʎ

Put x value in 1

-2(1-ʎ)+4+ʎ=0

3ʎ+2=0

ʎ=-2/3

Put ʎ value in 1

-2x+4-2/3=0 

2x=10/3 

x=5/3

Put x value in 3

5/3+4y-3=0

4y=4/3

y=1/3

x=5/3

y=1/3

We get x = 5/3 and y =1/3.

The value of the function at (5/3,1/3) is 4.

The constraint also gets satisfied with these values.

5/3+4*1/3=9/3=3