Question

Find the critical numbers of the function.
f(x) = x^4*e^-8x

Answer 1

Critical point:
A critical point at the interior of the domain of a function is the point where the derivative is zero or undefined.

The domain of the given function

f(x) = x^4*e^-8x

is [-∞∞], so you will need to find the critical points on that interval.

Since there are no discontinuities, nor vertical asymptotes, you only have to worry about the extrema where f'(x)=0.

Calculate f'(x), equate to zero and solve for x in f'(x)=0. These are the critical points.

Hint: I only find one critical point, and it is quite obvious.

Answer 2

I found one critical point, but there are two. The critical point I found is 0, but I can't find the second one.

Answer 3

Yes, you're right, there are two critical points, namely two points at which f'(x)=0.

I erred in the calculations because I took the wrong function to start with.

The derivative is:
f'(x)=-4x³(2*x-1)e^-8*x
which obviously has two roots. I am quite sure you would be able to spot both roots.