Science:Math Exam Resources/Courses/MATH104/December 2013/Question 02 (e)

MATH104 December 2013

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Question 02 (e)
Consider the function $f(x)={\frac {x^{2}}{x^{2}-4}}.$ Its first and second derivatives are given by $f'(x)=-{\frac {8x}{(x^{2}-4)^{2}}},\quad f''(x)={\frac {8(3x^{2}+4)}{(x^{2}-4)^{3}}}$ (e) Find the coordinates of all local maxima, local minima, and inflection points. Be sure to indicate which is which.

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
Refer to parts (a), parts (b) and parts (c) to help with this problem.

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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 parts (b), we can see that there's no inflection point. On the other hand, using parts (a) and parts (c) we can see that the extrema can only occur possible at $\displaystyle x=0$, $x=\pm 2$. Since the latter two points are not in the domain, we need only to check the point $\displaystyle x=0$ for extrema. Looking at parts (c), we see to the immediate left of 0, the function is increasing and to its immediate right the function is decreasing. Thus this point is a local maximum. The coordinates are given by $\displaystyle (0,0)$ found by plugging in 0 into the original function.

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