Applications of Quadratic Functions

Question

According to data from the U.S Census Bureau, the population ( in thousands) of Detroit, Michigan, in year x can be approximated by g(x) = -.417x^2 + 48.6x + 250, where x=0 corresponds to 1900. In what year did Detroit have its largest population?

Answer 1

To find a maximum/minimum, take the first derivative of the function and set it equal to zero. That will tell you where the extreme point is, but you need to take the second derivative and see if it's > 0 (minimum), < 0 (maximum) or = 0 (saddle point) at that point.

 

The first derivative is

 

dg/dx = -0.834x + 48.6 = 0

 

Solving for x gives x = 48.6 / 0.834 = 58.3, which corresponds to ~1958. This is probably the correct answer, but taking the second derivative just to be sure gives

 

d²g/dx² = -0.834 < 0

 

This is a maximum, as expected.