$\"\"$

(a)

The logistic differential equation (4) : $\"\"$ .

$\"\"$

Implicitly differentiate with respect to $\"\"$ .

$\"\"$

Substitute $\"\"$ in the above equation.

$\"\"$

(b)

The population will be growing fastest when $\"\"$ reaches to maximum.

To find the maximum value equate $\"\"$ to zero.

$\"\"$

$\"\"$ , $\"\"$ and $\"\"$ .

$\"\"$ , $\"\"$ population never grows.

$\"\"$

So population is growing fastest when it reaches half its carrying capacity.

$\"\"$

(a) $\"\"$ .

(b) So population is growing fastest when it reaches half its carrying capacity.