Check if a utility function represents a monotone preference Given a function $u(x_1, x_2) = x_1 +x_2 + \min(2x_1, x_2)$, how do we mathematically prove that it monotonic or not? 

Is there is a general algebraic technique to show monotonicity of suchlike functions?
Using this definition of monotone preferences (Thanks to User Art). Let $X\subseteq \mathbb{R}^n$ be the set of possible consumption bundles, and $x,y\in X$. We have weak monotone preferences if $y&gt;&gt;x \implies y\succ x $, and strict preferences if $y\geq x \implies y\succ x$. If we have a utility function $u: X\to \mathbb{R}$, then $y\succ x$ is the same as $u(y)&gt; u(x)$. 

Regarding notation for two bundles $x,y\in X, x=(x_1,x_2,...,x_n), y=(y_1,y_2,...,y_n)$, we say $y&gt;&gt;x$ if $y_i&gt;x_i$ forall $i=1,2,...,n$ and we say $y\geq x$ if $y_i\geq x_i$ forall $i=1,2,...,n$ and there is a $j$, such that $y_j&gt;x_j$. (This notation is not universal) 

Now, see if you can use this to answer your question