Science:Math Exam Resources/Courses/MATH110/April 2012/Question 05 (e)

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MATH110 April 2012

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Question 05 (e)
Let Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \displaystyle f(x)={\frac {2x^{2}}{x^{2}-1}}} (e) Determine where ƒ is increasing and where it is decreasing.

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
Whether f is increasing or decreasing is related to the sign of its first derivative, f' .

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Solution

We first find the derivative of f(x). Using the quotient rule, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle f'(x) = \frac{4x(x^2 - 1) - (2x)(2x^2)}{(x^2 - 1)^2} = \frac{4x^3 - 4x - 4x^3}{(x^2 - 1)^2} = \frac{-4x}{(x^2 - 1)^2}} . Since the denominator is always positive on the domain (squaring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle x^2 - 1} always yields a positive number), we only need check whether the numerator is positive or negative. When $x<0$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle -4x} is positive and when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle x > 0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle -4x} is negative. Thus f is increasing on the interval Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-\infty ,-1)\cup (-1,0)} and decreasing on the interval Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0,1)\cup (1,\infty )} .

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