Question

The number of mosquitoes M(x), in millions, in a certain area depends on the June rainfall x, in inches: M(x) = 4x - x2. What rainfall produces the maximum number of mosquitoes?

Answer 1

Conditions

The number of mosquitoes $M(x)$, in millions, in a certain area depends on the June rainfall $x$, in inches: $M(x) = 4x - x2$. What rainfall produces the maximum number of mosquitoes?

Solution

We must find the maximum of this function. For this consider the derivative:

$$f'(x) = (4x - x^2)' = 4 - 2x$$

The point $x=2$ is a minimum of this function, and as the rainfall in inches can't be negative value, so $x=0$ is a maximum, $4-2*0 = 4$ millions of mosquitoes.