Question

The percent of a​ country's households with broadband Internet access can be modeled by the function f left parenthesis x right parenthesisequalsnegative 0.11 x squared plus 6.51 x plus 3​, where f left parenthesis x right parenthesis is the projected percent of households with broadband Internet access and x is the number of years since 2000.

a. Will this function have a maximum or a​ minimum? How can you​ tell?

b. According to this​ model, in what year will the percent of homes with broadband Internet access be at its maximum or​ minimum?

c. What is the predicted​ maximum/minimum percent of homes with broadband Internet​ access?

Answer 1

a) The function has a maximum at x = 29,59

b) 99,32 %

Explanation:

f(x) = -0,11*x² + 6,51*x + 3

Then:

f´(x) = -0,22*x + 6,51

If f´(x) = 0 then - 0,22*x + 6,51 = 0

0,22*x = 6,51

x = 6,51/0,22 ⇒ x = 29,59

if we get the second derivative

f´´(x) = - 0,22 f´´(x) < 0 then we have a maximun at x = 29,59

a) the function has a maximum

b) 29,59 years after 2000 ( in 2030 )

c) f(x) = - 0,11*x² + 6,51*x + 3

f(29,59) = - (0,11)* (29,59)² + 6,51*29,59 + 3

f(29,59) = 99,32 %