Science:Math Exam Resources/Courses/MATH102/December 2011/Question 04 (vi)

MATH102 December 2011

  •  Q1 (a) i  •  Q1 (a) ii  •  Q1 (b) i  •  Q1 (b) ii  •  Q1 (b) iii  •  Q1 (c)  •  Q2 (a) i  •  Q2 (a) ii  •  Q2 (a) iii  •  Q2 (b) i  •  Q2 (b) ii  •  Q2 (b) iii  •  Q2 (c)  •  Q3  •  Q4 (i)  •  Q4 (ii)  •  Q4 (iii)  •  Q4 (iv)  •  Q4 (v)  •  Q4 (vi)  •  Q5  •  Q6  •  Q7 (i)  •  Q7 (ii)  •  Q7 (iii)  •  Q8 (i)  •  Q8 (ii)  •  Q8 (iii)  •  Q9  •  

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Question 04 (vi)
Consider the function $f(x)={\frac {1}{4}}x^{4}-2x^{3}+{\frac {9}{2}}x^{2}.$ vi) Provide a qualitatively accurate sketch of f(x). Make sure your graph reflects the information above.

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

The previous questions have given you all the information you need to be able to transfer all that algebraic information into something geometric (the sketch of the graph). Ask yourself: What does an asymptote (vertical or horizontal) look like? What does a local maximum or minimum look like? What about an inflexion point? What does it look like when the function is increasing/decreasing? What does it look like when the function is concave up/concave down? Whatever you sketch, check whether your drawing matches the information you collected.

Checking a solution serves two purposes: helping you if, after having used the hint, 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.

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. This function is a polynomial of degree 4, so the curve sketching shouldn't offer too many difficulties. We have collected the following information: The function has no horizontal asymptotes (from part i) ) The function has a global minimum at x = 0 (from part ii) ) The function has two inflection points at x = 1 and x = 3 (from part iii) ) The function is decreasing up to its minimum and then increasing, except at x = 3 where it has an inflexion point (from part iv) ) The function is concave up all the way up to x = 1, then concave down until x = 3 and then concave up again (from part v) ) To get a better idea of what to sketch, we quickly compute the second coordinates of the three points of interests: Global minimum at (0,0) Inflection points at (1, 11/4) and (3, 27/4) This allows us to sketch the graph of this function. As we saw above, this function is always positive, reaches a global minimum at the point (0,0) and has two inflection points. This should give you a picture that looks like:

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