<s>[INST] proof that a function is decreasing I need to prove that $$\sqrt{3+x^{1/3}}+2\over x-1$$ is a decreasing function in $(1,\infty)$ with the definition of a decreasing function; the problem I have is that this expression is difficult so please I need your help
 [/INST] $$\sqrt{3+x^{1/3}}+2\over x-1$$
Convince yourself that this function is decreasing on $(1,\infty)$ if and only if (with the substitution $x^3$ instead of $x$) the function
$$\sqrt{3+x}+2\over x^3-1$$
is decreasing. Convince yourself again that this is decreasing on $(1,\infty)$ if and only if (with the substitution $x^2+4x+1$ instead of $x$)
$${\sqrt{x^2+4x+4}+2\over (x^2+4x+1)^3-1} = \frac{1}{x((x^2+4x+1)^2+(x^2+4x+1)+1)}$$
is decreasing. But hey, the final form is obviously decreasing on $(1,\infty)$ (We have used $x^2+4x+4 = (x+2)^2$ and $x^3 - 1 = (x-1)(x^2+x+1)$ for the last step).
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