#### A population grows according to the logistic growth equation, dN/dt=rN(1-N/K)  where dN/dt is the rate of population growth, r is the intrinsic rate of increase, N is population size and K Is the carrying capacity of the environment.According to this equation, population growth rate is maximum atOption: 1 K/4Option: 2 K/2Option: 3 KOption: 4 2K

According to the logistic growth equation, the population growth rate is given by $\mathrm{\frac{dN}{dt}=rN(1-\frac{N}{K})}$, where dN/dt represents the rate of population growth, r is the intrinsic rate of increase, N is the population size, and K is the carrying capacity of the environment.

To determine the point at which the population growth rate is maximum, we can find the value of N that maximises the expression $\mathrm{rN(1 - \frac{N}{K})}$.

To find the maximum, we can take the derivative of the expression with respect to N and set it equal to zero:

$\mathrm{\frac{d}{dN}(rN(1-\frac{N}{K}))=r(1-2\frac{N}{K})=0}$

Solving for N, we get :

$\mathrm{1-2\frac{N}{K}=0}$

$\mathrm{2\frac{N}{K}=1}$

$\mathrm{2=\frac{K}{2}}$

Therefore, the population growth rate is maximum at $\mathrm{N=\frac{K}{2}}$.

The correct answer is option 2 that is $\mathrm{\frac{K}{2}}$