Science:Math Exam Resources/Courses/MATH110/April 2011/Question 01 (c)

MATH110 April 2011

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Question 01 (c)
Determine whether this statement is true. If it is, explain why; if not, give a counterexample. A point on a function may be both an infection point and a critical point.

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
Think about the definition of a critical point and an inflection point. Can both of these criteria occur at the same time?

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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. An inflection point is a point where $f''$ is zero or undefined and the graph changes concavity. A critical point is a point c where either $f'(c)=0$ or $f'(c)$ does not exist. It is possible for a point to satisfy both definitions. Such a function has a point where both the first and second derivative are equal to zero. One such simple function is $f(x)=x^{3}$. It has a horizontal tangent line at $x=0$ (making $x=0$ a critical point) and also changes concavity at $x=0$. Thus the statement is true.

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