Find the max&min of the following function :
subject to the constraint :
By the Lagrang Multipliers' method :
........(1)
.......(2)
.......(3)
(1) will be :
(2) will be :
(3) will be :
then ?!!
By using the method of Lagrange, we will get:
... (1)
... (2)
... (3)
... (4)
By substituting (1) in (2), we will get the equation :
..
when x=0 ---> y=0 , substitute this in 4 --->
so the first two points are and
when ---> z=4 from (3) ---> y=x from (1) and (2)
So when we have y=x and z=4 , By substituting this in (4) :
which don't have any solutions
so when , we do not have any points.
when ---> y=-4x from (1) and z=-1 from (3)
by substituting this in (4) --->
when x=1 --> y=-4 ---> third point is (1,-4,-1)
when x=-1 ---> y=4 ---> fourth point is (-1,4,-1)
The points are : and
Find the value of the function at these point, biggest --> max, smallest --> min.