Find all local extremums of $f(x)=x^{2}e^{-x}$ and decide if these are global extremums

Question

As all my other questions, this one isn't homework (it's preparation for an exam).

I'd like to know if I did everything correctly. In my previous task, I had a mistake in the first derivation. But for this task, I think everything is alright.

Find all local extremums of the following function and decide if these are global extremums (i.e. maxima or rather minima of the function on its entire domain) or not:

$f:\mathbb{R}\ni x\mapsto x^{2}e^{-x} \in \mathbb{R}$

I have started by checking the behavior of $f$ towards $+- \infty$ to see it more clear / see at all if the extremums are global or local. And for finding the limit, I had to use L'Hôpitals rule twice (not shown to keep it short).

$$\lim_{x\rightarrow+\infty}f(x)=0$$

$$\lim_{x\rightarrow-\infty}f(x)=\infty$$

This already tells me that on the left (negative) side of the coordinate system, there will be endlessly extremums. $\Rightarrow$ The function cannot have any global extremums.

Now derivate $f$ and calculate the extremum:

$$f'(x)=e^{-x}(2x-x^{2})$$

$\Rightarrow$

$$f'(x)=0$$

$$0=\frac{1}{e^x}(2x-x^{2})$$

$$0=2x-x^{2}$$

$$x^{2}=2x$$

$$x=2$$

Now use the $2^{nd}$ derivation to check if this is a maximum or minimum. It's a local maximum in point $P(2|\frac{4}{e^{2}})$.

Answer 1

You are almost right. But at the $0=2x-x^{2}$ step, you cannot conclude that $x=2$. Indeed, $0=2x-x^{2}=x(2-x)\implies x=0,2$. Thus, you must also check $x=0$.

Answer 2

From your penultimate line you should also deduce that $x=0$ which gives the minimum of zero

Answer 3

Your first conclusion that "This already tells me that on the left (negative) side of the coordinate system, there will be endlessly extremums." is wrong, in general. A simple counter-example is $f(x)=e^{-x}$ which possesses the limits mentioned in the question but does not have endlessly extrema.

Moreover in the second derivation, you missed $x=0$ which will give you the minimum when you calculate the second derivative.