Global max of a multivar func when restriced to a given plane.

[TylerH]

On this year's Virginia Tech test there was a problem to prove:
$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}\leq \frac{3\sqrt{3}}{2}$
when
$x+y+z=xyz$

So my approach was to consider the function, f, where
$f(x, y, z)=\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}$
then restrict f to the plane x+y+z=xyz, then maximize f.

How could I restrict f to that plane and then maximize?

 

[slider142]

The solution set of x + y + z = xyz is not a 2-dimensional plane. It is a level set of the function g(x, y, z) = x + y + z - xyz. Let's call that set of points M. To optimize a function subject to a constraint of this type, it is best to use the method of Lagrange multipliers. In particular, if P = (a, b, c) is a stationary point for f over the manifold M, then the gradient of f at P must be a scalar multiple λ of the gradient of g at P. This equation, plus the fact that g(P) = 0 provides a system of 4 equations in 4 unknowns that you can solve for the requisite points. You can then use the Hessian to determine the identity of these points.

 

[TylerH]

The solution set of x + y + z = xyz is not a 2-dimensional plane. It is a level set of the function g(x, y, z) = x + y + z - xyz. Let's call that set of points M. To optimize a function subject to a constraint of this type, it is best to use the method of Lagrange multipliers. In particular, if P = (a, b, c) is a stationary point for f over the manifold M, then the gradient of f at P must be a scalar multiple λ of the gradient of g at P. This equation, plus the fact that g(P) = 0 provides a system of 4 equations in 4 unknowns that you can solve for the requisite points. You can then use the Hessian to determine the identity of these points.

Oh! That's actually a sort of natural extension to the 3d stuff I learned in calc 3. Thanks!