Question

Find all values of x where the function f(x)=-4x^2/e^-4x has a relative maximum. If there are no such values of x, submit an empty answer.

Answer 1

To do this we will first calculate the derivative :

$\frac{d}{dx}(\frac{-4x^2}{e^{-4x}})=\frac{d}{dx}(-4x^2e^{4x}) = -8xe^{4x}-16x^2e^{4x}=-8x(1+2x)e^{4x}$

The roots are $x_1 = 0, x_2 = -\frac{1}{2}$ .

Now we will analize the sign of $f'(x)$ near these points :

At $x_1=0, f'$ changes it's sign from $+$ to $-$ (exponent is always positive, $(1+2x)$ remains positive around $x_1$), thus $x_1$ is a local maximum (and we can even see that it is a global maximum).

At $x_2=-\frac{1}{2}, f'$ changes it's sign from $-$ to $+$ ($x$ remains negative around $x_2$), so $x_2$ is a local minimum and therefore is not a local maximum.