Logistic equation showing $L/2$ has the most rapid increase on the interval $0So the question goes as follows: 
"Show that a solution $y(t)$ of the logistic equation having  $0<y(0)<L$ is increasing most rapidly when its value is $L/2$.(Hint: You do not need to use the formula for the solution to see this)"
So first of all I don't unnderstand how you would be able to see this without utilizing the formula:
$\frac{dy}{dt}=ky(1-\frac{y}{L})$
And also most important how would you see that the increase is as most rapid att $t=L/2$?
I was thinking about taking the difference between the slope at $t=L/2$ and the other $t$ values on the interval. But it does't seem right.
Any suggestions?
Thank you in advance!

Notice that we are only concerned about $$0<y<L$$ Since $$y'=ky(1-\frac{y}{L})>0$$ we have $$y''=ky'(1-\frac{2y}{L})$$  and $$y'''=\frac {-2k}{L}y'^2<0$$ Notice that the only critical point of $y'$ is at $$y=\frac {L}{2}$$This critical point is a maximum because $y'$ is concave down due to the fact that its second derivative, $y'''$ is negative.
