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

MATH110 April 2016

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Question 05 (i)
Let $f(x)={\frac {x-1}{x^{2}}}$. (i) Make a large sketch of the graph of the function $f$. Make sure you identify the $x$-coordinate of any intercepts as well as local extreme values and inflection points, if they exist.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Based on what we attained in last few questions, e.g. increasing, decreasing intervals; concave up and concave down, inflection, etc. we can draw a rough graph.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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.