Answer to Question #314226 in Calculus for pavan

Question #314226

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

1
Expert's answer
2022-03-20T06:44:03-0400

"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,\n2z+\\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"


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