00:01
Hello, let's have a look at the question.
00:04
So, an optimization problem is defined by the following objective function.
00:09
We have minimize f x y z is equals to 1 by 2 multiplied with x square plus y square and plus z square and it is subjected to g1 x y z that is equals to x minus y equals to 0 and we have g2 x y z that is equals to x plus y plus z minus 1 is equals to 0.
00:45
Now, here we have to use the lagrange multiplier method to find the solution for x and in the optimization problem.
00:54
So, we have that del f of x y z is equals to x y and z and we have g1 x y z is equals to x minus y equals to 0 and we can write that del g1 of x y z will be equals to 1 minus 1 and 0.
01:25
Similarly, we have g2 x y z is equals to x plus y plus z minus 1 equals to 0.
01:34
So, we have del g2 of x y z is equals to 1 1.
01:43
Now, here we will consider the system that del f x y z is equals to lambda multiplied with del g1 x y z and it is added to mu delta g2 and here we have x y z.
02:08
So, as we have x minus y equals to 0 and x plus y plus z minus 1 equals to 0.
02:16
So, here we can write that x y z is equals to lambda multiplied with 1 minus 1 and 0 and plus mu multiplied with 1 1 and 1.
02:30
So, we can further write this as lambda plus mu comma minus lambda plus mu and we have mu.
02:41
Also, this can be written as x is equals to lambda plus mu y is equals to minus lambda plus mu and z is equals to mu.
02:55
Now, let it be the a part.
02:57
Now, we had that x minus y is equals to 0.
03:01
So, from here we can say x is equals to y.
03:05
So, we can say now that this will be lambda plus mu is equals to minus lambda plus mu.
03:15
So, this will be equals to 2 lambda is equals to 0.
03:21
So, the value for lambda is equals to 0.
03:25
Also, we have that 2 x is equals to 1 minus z...