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,

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

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 $(-\mathrm{\infty },-1)\cup (-1,0)$ and decreasing on the interval $(0,1)\cup (1,\mathrm{\infty })$.