The number of mosquitoes, M(x), in millions, in a certain area depends on the June rainfall, x, in inches. The function that models the phenomenon is LaTeX: M\left(x\right)=16x-x^2M ( x ) = 16 x − x 2. Find the amount of rainfall that will maximize the number of mosquitoes. What is the maximum number of mosquitoes? (Hint: find the vertex

8 inches of rainfall will maximize the number of mosquitoes.

Maximum number of mosquitoes = 64 millions

Step-by-step explanation:

We are given the following in the question:

where M(x) is the number of mosquitoes in millions and x is the rainfall in inches.

First, we differentiate M(x) with respect to x, to get,

Equating the first derivative to zero, we get,

Again differentiation M(x), with respect to x, we get,

Thus, by double differentiation test the maxima occurs at x = 8 for M(x).

Thus, maximum number of mosquitoes are there when the rainfall is 8 inches.

Maximum number of mosquito:

Thus, maximum number of mosquito is 64 millions.

Vertex: (8,64)

The attached image shows the graph.

step-by-step explanation:

first, eliminate the parenthesis:

3

x

+

2

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4

x

second, group like terms:

3

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−

4

x

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2

third, combine like terms:

(

3

−

4

)

x

+

2

→

−

1

x

+

2

→

−

x

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2