The number of mosquitoes in anchorage, alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by: m (x) = - x^2+14xm (x) = - x 2 + 14xm, left parenthesis, x, right parenthesis, equals, minus, x, start superscript, 2, end superscript, plus, 14, x how many centimeters of rain will produce the maximum number of mosquitoes?

M (x) = - x ^ 2 + 14x

To find the maximum function, the first thing to do is to obtain the first derivative:

m ' (x) = - 2x + 14

We match zero:

0 = - 2x + 14

We clear x:

x = 14/2 = 7

x = 7

We verify that this value is a maximum through the second derivative:

m '' (x) = - 2

m '' (7) = - 2 (<0 therefore it is a maximum)

Answer:

7 centimeters of rain will produce the maximum number of mosquitoes.