Absolute Min and Max

Question

find the absolute maximum and absolute minimum values of f(x)=x-ln x on the interval [ 1/4,4]

1. Find the absolute maximum of f on the interval
2. Find the absolute minimum of f on the interval

Answer 1

To find the absolute maximum and absolute minimum values of a function f(x) on the given interval, we find the critical points, where the f’(x)=0 or does not exist (DNE), and then compare the values of the function at the ends of the interval and in the critical points.

1.      f’(x)=1-1/x

2.     f’(x) = 0 or DNE

               1- 1/x = 0 

              1/x = 1

              x=1 - critical point

              1 – 1/x exists for all values of x on the interval [1/4, 4]

3.     Values of f(x) :

End points of the interval

f(1/4) = ¼ - ln(1/4) = ¼ +ln(4) = 1.6362

f(4) = 4 – ln(4) = 2.6137         <- highest value

Critical point

f(1) = 1 – ln(1) = 1          <- lowest value

4.     The absolute maximum  value is  4 – ln(4) = 2.6137.

The absolute minimum value is 1.

Hope it helps.