Science:Math Exam Resources/Courses/MATH104/December 2012/Question 02 (c)

MATH104 December 2012

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Question 02 (c)
Consider the function $f(x)={\frac {2x^{3}}{3x^{2}-9}}$ Its first and second derivatives are given by $f'(x)={\frac {2x^{2}(x^{2}-9)}{3(x^{2}-3)^{2}}},\qquad {}f''(x)={\frac {4x(x^{2}+9)}{(x^{2}-3)^{3}}}$ Find the x coordinate 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
Remember that local extrema can only occur at the points in the domain where the derivative is zero or undefined and the function swaps from increasing to decreasing or vice versa. Inflection points are points in the domain where the second derivative is zero or undefined and the concavity changes.

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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. By part(a) and part(b) of this problem we see that the function changes from increasing to decreasing at $x=-3$, so there is a local maximum there, and the function changes from decreasing to increasing at $x=3$, so there is a local minimum at x=3. The function changes concavity at $x=0$, so there is an inflection point at x = 0. (Notice that $x=\pm {\sqrt {3}}$ are not points in the domain and hence cannot be extrema or inflection points.)

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