# A certain population is known to be growing at a rate given by the logistic equation dx/dt=x(b-ax) Show that the minimum rate of growth will occur when occur when the population is equal to half the equilibrium size,that is,when the population is b/2a?

Jul 1, 2018

To optimise growth rate, take its derivative and set to zero:

$x ' = x \left(b - a x\right) = x b - a {x}^{2}$

$\therefore q \quad x ' ' = b - 2 a x = 0 q \quad \implies \boldsymbol{x = \frac{b}{2 a}}$

The next derivative confirms the nature of the critical point:

${x}^{\left(3\right)} = - 2 a < 0 q \quad a > 0$