Science:Math Exam Resources/Courses/MATH100/December 2013/Question 08 (b)/Solution 1

Where $f^{\prime }(x)>0$, $f(x)$ is increasing; where $f^{\prime }(x)<0$, $f(x)$ is decreasing.

From part (a), we note that $f^{\prime }(x)$ may change sign at its critical points $x=0$ and $x=1$. We can construct a sign table to organize our calculations:

	$(-\mathrm{\infty },0)$	$(0,1)$	$(1,\mathrm{\infty })$
$18x^{-\frac{5}{7}}$	$-$	$+$	$+$
$1-x$	$+$	$+$	$-$
$f^{\prime }(x)=(18x^{-\frac{5}{7}})(1-x)$	$-$	$+$	$-$

From this table we observe that $f(x)$ is increasing on $x\in (0,1)$ and decreasing on $x\in (-\mathrm{\infty },0)\cup (1,\mathrm{\infty })$.