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), $\underset{x\to \pm \mathrm{\infty }}{\lim }f(x)=0$ and vertical asymptote: $x=0$.

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

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

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

answer-graph