Constraint minimization by lagrange multipliers

Aim:-To minimize a given function using Lagrange Multipliers

Governing Equation:- `f=5-(x-2)^2-2(y-1)^2`

Objective:- The main objective of this challenge is to minimize the given function by the equation given using Lagrange multiplier that we are calling constraint minimization

Brief Theory:-

Lagrange Multiplier:- It is a method to find local maxima and minima of a given function subjected to equality constraints.

This method allows for solving the optimization without explicit parametrization.

For single constarint,

L(x,y,`lambda`)=f(x,y)-`lambda`g(x,y)

Brief Content of this project:-

This is the challenge for constraint minimization.

Given,

f(x,y)=`f=5-(x-2)^2-2(y-1)^2`

g(x,y)=x+4y-3

L(x,y,`lambda`)=`5-(x-2)^2-2(y-1)^2`-`lambda(x+4)`

Now,`(delL)/(delx)=0`

or, -2x+4+`lambda`=0 ..............(1)

`(delL)/(delx)=0`

or, -y+`lambda`+1=0...............(2)

`(delL)/(dellambda)=0`

or, x+4y=0.....................(3)

From 1 and 2,

-2x+4+`lambda`+y-`lambda`-1=`

Now, x+4(2x-3)-3=0

or, 9x-15=0

or, x=5/3

y=2(5/3)-3=1/3

`lambda`=-2/3

These values minimizes the function.