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Homework Statement
Another model for a growth function for a limited pupulation is given by the Gompertz function, which is a solution of the differential equation
dP/dt = c*ln(K/P)*P
where 'c' is a constant and 'K' is the carrying capacity.
At what value of P does P grow fastest?
Homework Equations
c = .05
K=1000
P_0 = 500 (initial condition)
P(t) = 1000/e^(e^(-.05t-.3665)) (this is the specific solution)
The Attempt at a Solution
I think it is asking to find what the max value of dP/dt. However I think to do this, I need to find the derivative of dP/dt which is d^2P/dt^2 and set it equal to zero. This will let me find t when the slope of P is at its max and then I plug t back into P. Is this correct?
If it is, I am in for one hell of a derivative...
Thanks!