Find the minimum value of fx y z x2 y2 z2 5 subject to x

Find the minimum value of

f(x, y, z) = x² + y² + z² 5

subject to x = y.

find f_min =

Also find the corresponding point

(x, y, z).=

Solution

f(x, y, z) = x² + y² + z² 5

x=y

let g(x,y,z)= x -y

by lagrange multipliers

f=g

<2x,2y,2z>=<1,-1,0>

2x= ,2y=- ,2z=0

x=/2 ,y=-/2 , z=0

we have x=y

/2 =-/2

/2 +/2 =1

=1

x=1/2 ,y=-1/2 ,z =0

f min= f(1/2 ,-1/2 ,0)= (1/2)² + (-1/2)² + 0² 5

f min= f(1/2 ,-1/2 ,0)= -4.5

corresponding point (x, y, z).=(1/2 ,-1/2 ,0)