How to check if the given function/surface has multiple local max or min?

Question

Consider the following function of $x_1$, $x_2$ and $x_3$ that each lie in $\mathbb{R}^1$: $$\frac{-10x_1+10x_2-x_3}{x_1^2+x_2^2+x_3^2}$$

How do I check if this function has multiple local max or min? It's clear that when $x_1=x_2=x_3=0$, then the function is not defined. Fixing two of the $x_{-i}$ equal to 0 and letting $x_i$ approach 0 makes the function unbounded. Depending on the direction we approach from, the function will either go to positive infinity or negative infinity. What I am interested in knowing is that does this function have any local minimum or maximum? Further, what about the more general case where we have $x_i$ and $i=1,\dots,N$ and the number we multiply each $x_i$ with in the numerator is given/fixed and known.

Answer 1

Hint.

Calling $X = (x_1,x_2,x_3)$ and $\vec v = (a,b,c)$ we have

$$ f(X) = \frac{X\cdot \vec v}{||X||^2} $$

so the relative minima/maxima follow the condition $X = \lambda \vec v$ or

$$ f(\lambda) = \frac{1}{\lambda} $$

Answer 2

For simplicity, let's rename $(x_1,x_2,x_3) \to (x,y,z)$ and consider the resulting function $$f(x,y,z) = \frac{10y-10x-z}{x^2+y^2+z^2},$$ looking for its local optima.

The candidates for local optima will be given by the equation $\vec{\nabla} f = \vec{0}$, which results in 3 equations: $$ 0 = \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} = \frac{\partial f}{\partial z}. $$ Can you compute the partial derivatives and finish solving the system?