Hello!

I'm new to the forum, so please excuse any mistakes I might be making.

I'm having trouble solving an exercise, I feel like there's something obvious I'm missing but can't see where...

There's a

function which is continuous everywhere and \[\lim_{x\to-\infty}f(x)\times\lim_{x\to+\infty}f(x)<0\] (both limits exist and are finite. I'm supposed to find the maximum of \[g(x)=\dfrac{1}{1+\left[f(x)\right]^2}\]

Since those limits have different signs, exist and are finite, I believe f has maximum and minimum (given by those two asymptotes)...

I got to the derivative of g easily, but I'm not sure what it gets me. \[g'(x)=\dfrac{-2f(x)f'(x)}{\left(1+\left(f(x)\right)^2\right)}\] but since I don't know much about f, I can't say anything about the sign of g'... I've tried to find the second derivative, but in one attempt it was always positive, so it couldn't have a maximum...

Can anyone help me see what I'm missing? Thanks in advance!

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