A differentiable function is defined on all real numbers

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A differentiable function

g

is defined on all real numbers. The signs of its first and second derivatives on four consecutive intervals are given in the table.

Interval	$(-\infty,-2)$	$(-2,1)$	$(1,4)$	$(4,\infty)$
$g'(x)$	$+$	$+$	$-$	$-$
$g''(x)$	$-$	$+$	$+$	$-$

Answer the following, showing clear reasoning that refers to the information in the table. (a) Determine the open intervals on which

g

is increasing. (b) Determine the open intervals on which

g

is concave down. (c) Determine the open intervals on which

g

is simultaneously increasing and concave up. (d) Classify the points

x=-2,\,1,\,4

as locations of relative extrema, points of inflection, both, or neither. Justify each classification using derivative signs.