Answer:
The number of mosquitoes is maximum for 6.5 inches rainfall.
Step-by-step explanation:
The given function is
[tex]M(x)=13x-x^2[/tex] .... (1)
where, M(x) is number of mosquitoes in millions and x the June rainfall in inches.
We need to find the rainfall that produces the maximum number of mosquitoes.
Differential the above function with respect to x.
[tex]M'(x)=13-2x[/tex] .... (2)
Equate first derivative equal to 0.
[tex]13-2x=0[/tex]
[tex]2x=13[/tex]
[tex]x=\frac{13}{2}=6.5[/tex]
Differential function (2) with respect to x.
[tex]M''(x)=-2<0[/tex]
Double derivative is negative. So, the value of function is maximum at x=6.5.
Therefore, the number of mosquitoes is maximum for 6.5 inches rainfall.
