Question

.Find the minimum value of with the constraints `xy+yz+zx=3

Answer 1

$Let F=(x^2+y^2+z^2) +\lambda(xy+yz+zx)=0$

$F_x=0, F_y=0, F_z=0$

$2x+\lambda(y+z)=0,2y+\lambda(x+z)=0, 2z+\lambda(x+y)=0$

$\lambda=\frac{-2x}{y+z},\lambda=\frac{-2y}{x+z},\lambda=\frac{-2z}{x+y}$

Equating the equations ,we have,

$\frac{-2x}{y+z},=\frac{-2y}{x+z},=\frac{-2z}{x+y}$

Taking,

$\frac{x}{y+z},=\frac{y}{x+z}$

We have, $(x-y)(x+y+z)=0$

$x=y or x+y+z=0$

we must have x=y since x+y+z cannot be 0

Suppose x+y+z=0, squaring this we have,

$(x^2+y^2+z^2)+2(xy+yz+zx)=0$