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

MATH104 December 2013

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Question 02 (a)
Consider the function $f(x)={\frac {x^{2}}{x^{2}-4}}.$ Its first and second derivatives are given by $f'(x)=-{\frac {8x}{(x^{2}-4)^{2}}},\quad f''(x)={\frac {8(3x^{2}+4)}{(x^{2}-4)^{3}}}$ (a) Find all $x$ such that $f'(x)=0$ or $f'(x)$ does not 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
Recall that a fraction is 0 or undefined if either the numerator or the denominator is equal to 0.

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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. Setting the numerator and the denominator of the first derivative to 0 gives $\displaystyle -8x=0$ and so $\displaystyle x=0$ $\displaystyle (x^{2}-4)^{2}=0$ and so $\displaystyle 0=x^{2}-4=(x-2)(x+2)$ giving solutions $\displaystyle x=\pm 2$.

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