Mathematics

Problem

Let

f:(0,\infty) \to \mathbb{R}

be given by

f(x)=2\ln x - x + 3

. a. Part 1: (i) Show that

f

is continuous on

(0,\infty)

and evaluate

\lim_{x\to 0^{+}} f(x)

and

\lim_{x\to \infty} f(x)

. b. Part 2: (ii) Determine the critical point(s) of

f

and discuss its monotonic behaviour. c. Part 3: (iii) Using the above information, find the range of

f

and hence decide whether

f

is surjective onto

\mathbb{R}