$\"\"$

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(a)

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

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$\"\"$

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

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$\"\"$

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

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$\"\"$

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$\"\"$

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(b)

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

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

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$\"\"$

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

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

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

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

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$\"\"$

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$\"\"$

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

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$\"\"$

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(a) $\"\"$ .

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