How can one show that this is positive?

Question

My simulation suggests that the following is positive for $x>0$.

$$f(x)=e^x-\dfrac{e^x}{x}+\dfrac{e^x}{x^2}-\dfrac{1}{x^2}$$

How one can prove that it is positive? Can anyone help?

Answer 1

Follow dxiv's suggestion. Note that since $x^2e^{-x}>0$, $f(x) > 0$ if and only if $$g(x):=x^2e^{-x}f(x) = x^2-x+1-e^{-x}>0.$$ Now notice that $g(0)=0$, so to show $g(x)>0$, it suffices to show that $g$ is increasing for $x>0$, i.e., that $g'(x)>0$ for $x>0$.

Then in order to show $g'(x)= 2x-1+e^{-x}>0$ we repeat this process. Note that $g'(0)=0$, so again we just need to show that $g''(x)>0$. Now this is ok, since $g''(x)=2-e^{-x}$, and $e^{-x}<1$ for $x>0$, so $2-e^{-x}>0$ for $x>0$.

Thus $g(x)>0$ for $x>0$, and hence $f(x)>0$ for $x>0$.