Science:Math Exam Resources/Courses/MATH110/April 2012/Question 05 (e)/Solution 1

We first find the derivative of f(x). Using the quotient rule,

$f'(x)={\frac {4x(x^{2}-1)-(2x)(2x^{2})}{(x^{2}-1)^{2}}}={\frac {4x^{3}-4x-4x^{3}}{(x^{2}-1)^{2}}}={\frac {-4x}{(x^{2}-1)^{2}}}$.

Since the denominator is always positive on the domain (squaring $x^{2}-1$ always yields a positive number), we only need check whether the numerator is positive or negative. When $x<0$, $-4x$ is positive and when $x>0$, $-4x$ is negative. Thus f is increasing on the interval $(-\infty ,-1)\cup (-1,0)$ and decreasing on the interval $(0,1)\cup (1,\infty )$.