Science:Math Exam Resources/Courses/MATH100/December 2016/Question 07 (a)

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MATH100 December 2016

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Question 07 (a)
Let $f(x)$ be a continuous function on the open interval $(a,b)$ . Which of the following four statements are always true? Select all that apply. If $\lim _{x\to a+}f(x)=-\infty$ and $\lim _{x\to b-}f(x)=\infty$ , then there is at least one zero of $f(x)$ inside $(a,b)$ . If $f(x)$ has a local minimum at $c$ in $(a,b)$ , then $f'(c)=0$ . $f(x)$ must have both maximum and minimum inside $(a,b)$ . If $f''(c)>0$ for some point $c$ in $(a,b)$ , then $f(x)$ has a local minimum at $c$ .

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
Consider the intermediate value theorem.

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Solution
(A) True — by the intermediate value theorem. (B) False — the derivative of a function may not even exist at a local minimum! For example, consider $f(x)=\|x\|$ on $(-1,1).$ Although $f$ has its local minimum at $x=0,$ $f'(0)$ does not exist. (C) False — just take the function $f(x)=x$ on $(a,b),$ which has no minimum or maximum on $(a,b).$ (D) False — the derivative could be everywhere positive. For example, consider $f(x)=x^{2}$ on $(1,2).$ While $f''(x)=2>0$ for any point $x$ in $(1,2)$ , it doesn't have local minimum on this interval. This is because $f$ is strictly increasing on the interval, i.e., $f'(x)=2x>0.$ The answer: $\color {blue}{\text{(A)}}$