<s>[INST] For which $a$ and $b$ the function $\frac{ax+b}{a^2x+b^2}$ is increasing? 
For which $a$ and $b$ the function  $$\frac{ax+b}{a^2x+b^2}$$ is increasing?

I know that function is increasing if $x_1 > x_2 \implies f(x_1)>f(x_2)$ but how can I find $a$ and $b$ for which this statement will be true?
 [/INST] Divide both polynomials:
$$\frac{ax+b}{a^2x+b^2}=\frac{a}{a^2}\left(1+\frac{b/a-b^2/a^2}{x+b^2/a^2}\right)=\frac{a}{a^2}+\frac{{a/a^2(b/a-b^2/a^2)}}{x+{b^2/a^2}}$$
This is function $1/x$ scaled and moved. The scale is
$$k=\frac{a}{a^2}\left(\frac{b}{a}-\frac{b^2}{a^2}\right)$$
You know that $c/x$ is decreasing when $c>0$ and increasing when $c<0$. In your case $k<0$:
$$\frac{a}{a^2}\left(\frac{b}{a}-\frac{b^2}{a^2}\right)<0$$
$$a(ab-b^2)<0$$
$$a^2b-b^2a<0$$
$$a^2b<b^2a$$
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