Dgdrxrt

I have attempted this optimization (through differentiation) question from the exercises I've received through a self-improvement course. I got stuck while differentiating the equation which is supposed to be the exponential growth model for bacteria growth, so I'm unable to solve for the answer, I am hoping someone can help to guide me through this question as I was and still am not adept at maths.

Here is the question:
The bacteria population in a certain colony is given by

$f(t) = 1000t^2e^-t$

where $t$ is time in minutes and $t ge 0$

Find the time at which the population reaches a maximum.

I've immediately attempted the product rule to try and differentiate, but as I said I got stuck, and I can't solve for $t$ without differentiating (I can equate $fracdydx$ to 0 and then find the maximum). I'm also not sure where the information '$t$ is time in minutes' and $t ge 0$ comes into play when solving.

Please let me know of the steps needed to complete the question and where I've gone wrong, I would love to learn, thanks!

I have attached two screenshots of the full problem, and my attempt at another screenshot.

Problem Question

My attempted solution

I can't really follow your handwriting or solution. It says how to solve the problem in the question with a derivative test.

$$ f(t) = 1000t^2 e^-t tag1 $$

the product rule is given as

$$ (g cdot h)^' = g^' cdot h+ g cdot h^' tag2$$

the constant doesn't matter here, it simply scales so instead look at

$$ f_1(t) = g cdot h = t^2 e^-t tag3$$

then we have

$$ g(t) = t^2 , h(t) = e^-t implies g^'(t) = 2 t , h^'(t) = -e^-t tag4$$

which gives us

$$ f_1^'(t) = 2 t cdot e^-t - t^2e^-t tag5 $$

we rewrite this as

$$ f_t^'(t) = e^-tt(2 - t) tag6$$

Then we have

$$ t = 0, 2 $$
$$ f_1(2) = (2)^t e^-2 = 4 cdot e^-2 = frac4e^2 tag7 $$

so our maximum for $f(t)$ is given by

$$ f(2) = frac4000e^2 tag8 $$