Science:Math Exam Resources/Courses/MATH110/April 2013/Question 01 (a)

MATH110 April 2013

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Question 01 (a)
Determine whether the following statement is true or false. If it is true, provide justification. If it is false, provide a counterexample. (a) If ƒ is a polynomial of degree greater than 1, then ƒ has a tangent line that crosses the x-axis.

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
Assume that the implication of the statement is false. Is it possible that the assumption holds? In other words, what can you conclude if you assume that ƒ has no tangent line that crosses the x-axis?

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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. This statement is true. If ƒ has no tangent line that crosses the x-axis, then all tangent lines must have slope 0. Hence ƒ' = 0 at every point, and thus the polynomial ƒ is necessarily constant. This contradicts the assumption we started with. So the statement is true.

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