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)  •  

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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!

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?

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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.

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