Calculus Homework- Function & Absolute Max and Min

Question

Consider the function f(x)=2x2−2x+6, 0≤x≤6. The absolute maximum of f(x) (on the given interval) is _________ and the absolute minimum of f(x) (on the given interval) is __________.

Answer 1

f(x) = 2x² - 2x + 6   0 ≤ x ≤ 6

 

Since f(x) is continuous on [0,6], the absolute extrema occur either at a critical point or an endpoint.  

 

To find the critical points, solve the equation f'(x) = 0

 

                                                               4x - 2 = 0

 

                                                                     x = 1/2

 

1/2 is the only critical point on the interval

 

Check f(1/2), f(0), and f(6):

 

  f(1/2) = 5.5

 

  f(0) = 6

 

  f(6) = 66

 

Absolute max = f(6) = 66

Absolute min = f(1/2) = 5.5