Determine the maximum value of dP/dt where dP/dt = rP(1-(P/k)), r and k are positive constant, and interpret the meaning of this maximum value.

I do it like this,

let y = dP/dt

y=rP(1-(P/k))

dy/dP = r-2P(r/k)

(d^2)y/dP^2 = -2(r/k)

(d^2)y/dP^2 < 0 , so it is a maximum value.

When dy/dP = 0,

0 = r-2P(r/k)

P = k/2

Substitute P = k/2 into dP/dt = rP(1-(P/k)),

and I get dP/dt = rk/4

So the maximum value is (k/2 , rk/4)

Is that correct? I didn't use the differentiation for (d^2)P/dt^2 and (d^3)P/dt^3 because it is too complicated, can I really do like this? Please correct me if I am wrong and provide me a way to do it. Thanks in advance.