Science:Math Exam Resources/Courses/MATH104/December 2013/Question 01 (j)

MATH104 December 2013

• 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 • Q6 (a) • Q6 (b) • Q6 (c) •

Other MATH104 Exams

• December 2014 • December 2011 • December 2010 • December 2012 • December 2013 • December 2015 • December 2016 •

[hide] Question 01 (j)
If a function $f$ is continuous for all $x$ and if $f$ has a local maximum at (-1, 4) and a local minimum at (3, -2), which of the following statements must be true? (A) The graph of $f$ has an inflection point somewhere between $x=-1$ and $x=3.$ (B) $f'(-1)=0.$ (C) The graph of $f$ has a horizontal asymptote. (D) The graph of $f$ has a horizontal tangent line at $x=3.$ (E) The graph of $f$ intersects both axes.

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!

[show] Hint
Draw a picture with the two dots in it. Try to connect the dots. Which of the given condition appears to always be true? Can you justify it? Alternatively, can you create a picture where the other four conditions are false?

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.

[show] 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 (E). To show that E is correct, we can use the intermediate value theorem twice. In order for f to be a continuous function that connects $\displaystyle (-1,4)$ and $\displaystyle (3,-2)$ , we know that it must have some value at $\displaystyle x=0$ by the fact that the function is continuous for all real values of x. Thus, it crosses the y axis. As these two points live both above and below the x axis, we see that $\displaystyle f(-1)\geq 0\geq f(3)$ and since f is continuous, the intermediate value theorem states that there is some value c in the interval $\displaystyle (-1,3)$ such that $\displaystyle f(c)=0$ . Thus the answer is (E). Note. For the curious minded, the graph below gives a counter example to all of the other 4 conditions.