Sketch a graph that satisfies the following conditions using derivatives

Question

Plot the graph of a function that satisfies the following conditions:

• $\operatorname{dom}(f)=\Bbb{R}\setminus\{0\}$.
• $f'(x)<0$ in $(-\infty,0)$ and $(0,+\infty)$.
• $f''(x)>0$ in $(-\infty,-2)$ and $(0,+\infty)$.
• $f''(x)<0$ in $(-2,0)$.
• $\displaystyle\lim_{x\to0^-}f(x)=-\infty$.
• $\displaystyle\lim_{x\to0^+}f(x)=+\infty$.
• $\displaystyle\lim_{x\to-\infty}f(x)=+\infty$.
• $\displaystyle\lim_{x\to+\infty}f(x)=0$.

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I have thought about $f(x)=\dfrac{1}{x}$ and $g(x)=-e^{1/x}$ but clearly they do not fit in all conditions.

Answer 1

How's this look? I think this meets all the requirements. Don't try to come up with a description of a function...just try to draw the requirements

Answer 2

Instead of thinking of specific functions that could potentially meet these requirements, why not use the requirements as a guide to start drawing a function?

For example, you know $f(x)$ is decreasing on $(-\infty, 0)$ and that it is concave up from $(-\infty, -2)$ and concave down on $(-2,0)$. It also shoots off to $-\infty$ as you get close to $0$ from the left-hand side. Using these facts, try starting at a point in the third quadrant and sketch what $f(x)$ should look like as $x \rightarrow 0^-$.