MATH104 December 2015
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q7 • Q8 (a) • Q8 (b) • Q8 (c) • Q9 • Q10 (a) • Q10 (b) • Q10 (c) • Q10 (d) • Q11 •
Question 10 (b)
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Let . Its derivative satisfies
- .
(b) Find the intervals on which is increasing or decreasing.
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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?
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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.
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Hint 2
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Since we are only interested in the sign of and not its particular values, one of the given expressions for the derivative may be easier to work with than the others...
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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.
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Solution
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Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
To determine the intervals of increase/decrease of , we consider the sign of
Now may change sign at the zeroes of the numerator (which are the zeroes of itself; in this case ) or the denominator (in this case ). Let us therefore tabulate our results as follows:
Interval: |
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Sign of |
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Sign of |
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Sign of |
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Sign of |
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Considering the signs of each of the factors, we have
- for all (why?)
- when while when
- when and when
Hence we can complete our table:
Interval: |
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Sign of |
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Sign of |
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Sign of |
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Sign of |
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Since a (differentiable) function is increasing on an interval if on that interval (and similarly for decreasing), we have that is increasing on and decreasing on Note that we have excluded from the interval of increase as is undefined at
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