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

From part (b), no $y$-intercept and one $x$-intercept at $x=1$.

From part (c), $\lim _{x\to \pm \infty }f(x)=0$ and vertical asymptote: $x=0$.

From part (e), ${\textstyle f}$ is increasing in ${\textstyle (0,2)}$ and decreasing in ${\textstyle (-\infty ,0)\cup (2,\infty ).}$

From part (d), local maximum $(2,1/4)$ and no local minimum

From part (h), ${\textstyle f}$ is concave up in ${\textstyle (3,\infty )}$ and concave down in ${\textstyle (-\infty ,0)\cup (0,3)}$. Also, $f''(3)=0$. Since $f''$ changes its sign at $x=3$, the point $\left(3,{\frac {2}{9}}\right)$ is the inflection point. Based on these information, we draw the graph of $f$ as follows.