Science:Math Exam Resources/Courses/MATH104/December 2014/Question 05 (b)/Solution 1

Remark first that $f$ does not exist at $x=0$; hence, we have three intervals on which $f'(x)$ exists and is non-zero, namely $(-\infty ,0),(0,2),$ and $(2,\infty )$. We create a table with these intervals and the function that make up $f'$ to determine when $f'$ is positive or negative, hence when $f$ is increasing or decreasing.

	$(-\infty ,0)$	$(0,2)$	$(2,\infty )$
$e^{x}$	+	+	+
$(x-2)$	-	-	+
$x^{3}$	-	+	+
$f'$	+	-	+
$f$	increasing	decreasing	increasing

So $f$ is increasing on the intervals $(-\infty ,0)$ and $(2,\infty )$ and decreasing on $(0,2)$.