Because f(x) is continuous on the given interval, the EVT (Extreme Value Theorem) applies, and guarantees both an absolute max and an absolute min on this interval. The EVT specifies further that these extrema will occur at either of the endpoints of the given closed interval (in this case x = .1 and 2.1) or at critical points for f(x) (i.e. any x-values such that f'(x) = 0 or f'(x) undefined, the latter of which does not occur since f(x) differentiable on the closed interval).
Candidates for global and local extremes:
f(.1) = .2 - ln(.3) ~ 1.40
f(2.1) = 4.2 - ln(6.2) ~ 2.36
f'(x) = 2 - 1/x = 0
x = .5
f(.5) = 1 - ln(1.5) ~ .60
Thus, (2.1 , 2.36) is the global max , (.5 , .60) is the global min, and (.1 , 1.40) is a local max.
