Constraint Minimisation Problem

Problem:-

Minimize the function `5−(x−2)^2−2*(y−1)^2` subject to `x+4y-3=0`.

Solution:-

`f` = `5 − ( x − 2 )^2 − 2 * ( y − 1 )^2`

`g` = `x+4y-3`

According to Lagrange Multipliers,

`L = f + lamda*g`

So,

`L = 5- (x-2)^2 - 2*(y-1)^2 + lamda*(x+4y-3)`

`(del L)/(del x) = -2x +4 + lambda` `rarr` `Equation 1`

`(del L)/(del y) = -y +1 +lambda` `rarr` `Equation 2`

`(del L)/(del lambda) = x+4y -3` `rarr` `Equation 3`

Substracting `Equation 2` from `Equation 1`, we get-

`-2x + y +3=0` `rarr` `Equation 4`

Solving `Equation 3` and `Equation 4`,

Multiplying `Equation 3` by `2` and then adding it with `Equation 4`, we get

`y=1/3` and `x=5/3`

Substituting the values of `x and y` in `Equation 1`, we get-

`lambda= -2/3`

Hence `f` has a minimum subject to `g=0`