Science:Math Exam Resources/Courses/MATH100/December 2013/Question 08 (d)

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MATH100 December 2013

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Question 08 (d)
Let $\displaystyle y=f(x)=63x^{2/7}-14x^{9/7}$. Sketch the graph of $\displaystyle y=f(x)$ and indicate the inflection point(s) on your graph. (The results from parts (a), (b), and (c) may be useful.)

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
What are the critical numbers of $f$? What are the intervals of increase/decrease of $f$? How do these relate to the extrema of $f$?

Hint 2
What are the intervals of concavity of $f$? What is an inflection point, and how does it relate to concavity?

Hint 3
It may be useful to find the x-intercept(s) of $\displaystyle f$. In addition, note the behaviour of $\displaystyle f'$ near $x=0$.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

From parts (a), (b), and (c) we know the following about $f(x)=63x^{2/7}-14x^{9/7}$: The critical numbers of $f$ (where $f'$ is zero or does not exist) are $\displaystyle x=0,1$ $f$ is increasing on $x\in (0,1)$ and decreasing on $x\in (-\infty ,0)\cup (1,\infty )$ $f$ has a local minimum at $\displaystyle x=0$ and a local maximum at $\displaystyle x=1$ (note the change in the increase/decrease of $f$) $f$ is concave up on $x\in (-\infty ,-{\tfrac {5}{2}})$ and concave down on $x\in (-{\tfrac {5}{2}},0)\cup (0,\infty )$ $f$ has an inflection point (changes concavity) at $x=-{\tfrac {5}{2}}$ $\displaystyle f(x)=x^{\frac {2}{7}}(63-14x)$ and thus $f$ has x-intercepts at $x=0,{\tfrac {9}{2}}$ $\lim _{x\to 0^{-}}f'(x)=-\infty$ and $\lim _{x\to 0^{+}}f'(x)=\infty$, thus the graph of $f$ is very steep near $x=0$ Plotting the points of interest gives the following: Accounting for increase and decrease gives the following rough sketch: Finally, incorporating concavity leads us to the final graph: $\displaystyle f(x)=63x^{2/7}-14x^{9/7}$

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