00:01
Function is given as f is equals to x y z and upon that a condition is x square minus 4 plus 4 y square plus 4 z square is equals to 4.
00:15
We have to find maximum value and minimum value of f under this condition.
00:20
We will use lagrange multiple formula for this or we can say the lagrange multiple method for that capital f is equals to small f plus lambda times g.
00:31
What is g? we can shift this 4 to right side and this will come equation equals to 0 and that equation will be g.
00:37
If we put here value it will be x y z plus lambda times g.
00:42
G is x square plus 4 y square plus 4 z square minus 4 that is f.
00:51
For finding the critical point we get we will find f of x f of y and f of z.
00:57
If we find its value these values are x y if we derivative with respect to x we get y z plus lambda it become 2x.
01:13
If we derivative with respect to y we get x z plus lambda 2 to the 4 to the 8 y x y plus lambda times 8 z and for critical point this should be equals to 0.
01:31
From here if we multiply this by x we get x y z is equals to minus time 2 lambda x.
01:39
From here x y z if we multiply by y we get this from minus 8 lambda y square and from here x y z will be equals to minus 8 lambda z square.
01:56
We are saying that x y z are all equal here so x y z is equals to negative 2 lambda x square negative 8 is equals to negative 8 lambda y square is equals to negative 8 lambda z square.
02:18
We can say it as another constant k.
02:22
From here we get 2 x square is equals to minus k upon lambda and from here we get 8 y square is equals to minus k over lambda.
02:40
Similarly 8 z square is also minus k upon lambda.
02:46
Now from here what we do we will just put values here...