CONSTRAINT MINIMIZATION OF GIVEN EQUATION USING PEN AND PAPER METHOD:

5-(x-2)²-2(y-1)²

subject to the following constraint

x + 4y = 3

Here consider

SOLUTION:

consider f(x,y) = 5-(x-2)²-2(y-1)²

Condition h(x,y)= x + 4y = 3

By using lagrange: method

g(x,y)= f(x,y)+∆(h(x,y))

At minima condition

∂g/∂x = ∂/∂x( f(x,y)+∆(h(x,y)))

∂g/∂x= -2(x-2)+∆ = -2x+∆+4

∂g/∂y= -2×2(y-1)+4∆ = -4y+4∆+4

∂g/∂∆= x+4y-3

Equating

∂g/∂x= 0

i.e -2x+∆+4 =0

∂g/∂y= 0

i.e -4y+4∆+4 =0

∂g/∂∆= 0

i.e x+4y-3 =0

On solving

∆=-2/3

y=1/3

x=5/3

conclusion: hence we find out minimum values of x and y