Find the value of α so that the function has a stationary point

Question

The function is $f(x)=(x^2 + \alpha)e^{-5x}$

I tried to calculate the second derivative, but then I have no idea how to continue from there

Edit: Sorry about the formatting. No idea how to fix it on my phone

Answer 1

That's a correct way, we have

$$f(x)=(x^2 + α)e^{-5x}\implies f'(x)=(2x+\alpha)e^{-5x}-5(x^2 + α)e^{-5x}=0$$

and then, since $\forall x \in \mathbb R,\quad e^{-5x}>0$, we need

$$(2x+\alpha)-5(x^2 + α)=0$$

and we need to find $\alpha$ such that the quadratic has only one real root.