Step 1 :The number of mosquitoes, $M(x)$, in millions, in a certain area depends on the June rainfall, $x$, in inches. The function that models that phenomenon is $M(x)=12 x-x^{2}$. We need to find the amount of rainfall that will maximize the number of mosquitoes and the maximum number of mosquitoes.
Step 2 :The maximum or minimum of a quadratic function occurs at its vertex. The x-coordinate of the vertex of a function in the form $f(x)=ax^{2}+bx+c$ is given by $-\frac{b}{2a}$. In this case, $a=-1$ and $b=12$, so the x-coordinate of the vertex is $-\frac{12}{2(-1)}=6$. This means that the amount of rainfall that will maximize the number of mosquitoes is 6 inches.
Step 3 :To find the maximum number of mosquitoes, we substitute $x=6$ into the function $M(x)$.
Step 4 :Final Answer: The amount of rainfall that will maximize the number of mosquitoes is \(\boxed{6}\) inches. The maximum number of mosquitoes is \(\boxed{36}\) million.