Science:Math Exam Resources/Courses/MATH104/December 2011/Question 01 (k)

MATH104 December 2011

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q2 (e)  •  Q2 (f)  •  Q2 (g)  •  Q3  •  Q4  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q5 (d)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  Q6 (e)  •  

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Question 01 (k)

Let g be a continuous function on a close interval [0,1]. Let g(0) = 1 and g(1) = 0. Which of the following is NOT necessarily true? There exists a number c in [0,1] such that g(c) ≥ g(x) for all x in [0,1]. For all a and b in [0,1], if a = b then g(a) = g(b). There exists a number c in [0,1] such that g(c) = 1/2. There exists a number c in [0,1] such that g(c) = 3/2. For all c in the open interval (0,1), $\lim _{x\rightarrow c}g(x)=g(c)$ Note: the Dec 2011 exam had a typo in option E: g(c) in the limit should have been g(x)

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 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.

Hint 1
Try to think back to the theorems and definitions in this course and see which statements are satisfied by these theorems and definitions.

Hint 2
The following theorems and definitions might be useful: The intermediate value theorem, The extreme value theorem, The definition of continuity.

Hint 3

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.

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. The correct answer is (d). Let's examine each possibility. (a) Since $g(x)$ is continuous, we can use the extreme value theorem which says exactly what (a) is telling us - there is a maximum value on the interval $[0,1]$ (b) This is saying that if two values are equal then their function values are equal. Since a function can only take one output value for each input value, this is a true statement. (c) Since $g(0)=1>1/2>0=g(1)$ and $g(x)$ is continuous, then the intermediate value theorem tells us exactly what this statement says, there is a value c in $[0,1]$ such that $g(c)=1/2$ (d) This statement is false. The difference between (c) and this problem is that the number 3/2 does not lie between $g(0)$ and $g(1)$. A counter example would be a straight line joining $g(0)$ and $g(1)$. This can be described by $y=-x+1$. (e) This is true as this is the definition of continuity and $g(x)$ is given to be continuous.

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