Science:Math Exam Resources/Courses/MATH110/April 2016/Question 05 (h)/Solution 1

From the part (g) of this question, we know that

$$f''(x)={\frac {2x-6}{x^{4}}}.$$

Also, from the part (a), the domain of $f$ is $\{x\neq 0\}$. Since ${\textstyle x^{4}>0}$ when ${\textstyle x\neq 0}$, the sign of the second derivative $f''$ is determined by the sign of the numerator $2x-6$. Indeed, we can easily see that

$${\begin{aligned}2x-6&=2(x-3)>0{\text{ for }}x>3,\\2x-6&=2(x-3)<0{\text{ for }}x<3.\end{aligned}}$$

This implies that $f''>0$ when $x\in (3,\infty )$ and $f''<0$ when $x\in (-\infty ,0)\cup (0,3)$.

Therefore, ${\textstyle f}$ is concave up in ${\textstyle \color {blue}(3,\infty );}$ while ${\textstyle f}$ is concave down in ${\textstyle \color {blue}(-\infty ,0)\cup (0,3)}$.