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.
Best Answer
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
$$ f(\lambda) = \frac{1}{\lambda} $$