Mathematics

Problem

For a twice-differentiable function

f

, the values of its derivatives at three distinct points are recorded in the table below.

Point $x$	1	2	3
$f'(x)$	0	0	0
$f''(x)$	5	-2	0

a. Part 1: a) Classify each point

x=1,2,3

as corresponding to a local minimum, a local maximum, or an inconclusive case for

f

. Provide justification in each instance. b. Part 2: b) Additional information is supplied:

f(1)=4,\;f(2)=5,\;f(3)=2

, and

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

. Determine the absolute minimum value of

f

\mathbb{R}

and support your answer.