Find the absolute minimum and maximum values of f(x)=x−ln⁡x over the interval [1/2,2]

Question

Find the absolute minimum and maximum values of f(x)=x−ln⁡x over the interval [1/2,2]

Minimum value: 

Maximum value: 

Answer 1

f(x) = x - lnx

f'(x) = 1 -1/x = 0

x- 1 = 0

x=1 is a local extremum, an absolute & relative minimum

f(1) = 1-ln1 = 1-0=1

check trhe endpoints

f(2) = 2-lnx = about 2-.7 = 1.3= absolute max

f(1/2) = 1/2 -(-.7) = .5 +.7 + 1.2 < 1.3