<s>[INST] What is the smallest possible value of their sum? The product of two positive numbers is 36. What is the smallest possible value of their sum?
so far I got
$$xy=36$$
$$y=\frac{36}{x}$$
 [/INST] First note that
$$f(x)=x+\frac{36}{x}=x+36x^{-1}$$
$$\frac{d}{dx}[f(x)]=1-36x^{-2}=1-\frac{36}{x^2}$$
So now let's find the critical points
$$\frac{d}{dx}[f(x)]=0 $$
$$ 1-\frac{36}{x^2}=0 $$
$$ 1=\frac{36}{x^2} $$
$$ x^2=36= (\pm 6)^2 $$
$$ x=\sqrt{(\pm 6)^2}=\pm 6 $$
Since we're only considering positive values, we'll omit the negative value. Next we must check if $x=6$ is a local minima of $f(x)$. So now
$$\frac{d^2}{dx^2}[f(x)]=72x^{-3}=\frac{72}{x^3}$$
$$\frac{d^2}{dx^2}[f(6)]=\frac{72}{6^3}\gt 0 $$
Therefore, $f(x)=x+\frac{36}{x}$ has a local minima at $x=6$. Which implies that the smallest sum of positive values is 
$$f(6)=6+6=12$$
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