Solve the given optimization problem by using substitution. HINT [See Example 1.]

Find the minimum value of
f(x, y, z) = x^(2) + y^(2) + z^(2) − 7
subject to z = y.

Question

Solve the given optimization problem by using substitution. HINT [See Example 1.]

Find the minimum value of 
f(x, y, z) = x^(2) + y^(2) + z^(2) − 7
 subject to z = y.

ganeshie8 · Answer

use the hint ?

anonymous · Answer

doesn't really help

anonymous · Answer

its the same thing for every problem

ganeshie8 · Answer

substitute z=y in the given function,
that eliminates z from the function

ganeshie8 · Answer

f(x, y, z) = x^(2) + y^(2) + z^(2) − 7

put z = y

f(x, y) = x^2 + y^2 + y^2 - 7
         = x^2 + 2y^2 - 7

anonymous · Answer

so fx = 2x
fy = 4y

ganeshie8 · Answer

oh u wana use lagrange ? its a overkill for this problem..

ganeshie8 · Answer

think a bit, whats the minimum value of this ? can the function get any lower than  -7 ?

anonymous · Answer

no, so -7 is the min?

ganeshie8 · Answer

since x^2+2y^2 is always positive, the minimum value is 0 for this.
so, when x=y=z=0, the function has a minimum value of -7

ganeshie8 · Answer

yes

anonymous · Answer

ohh okay that makes sense

ganeshie8 · Answer

good, if u wana carry out lagrange... u may do that as well... 
guess ur professor wants u show that work

anonymous · Answer

yeah he taught us through that way but he hasn't done examples that are on the homework thats why I'm confused