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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!