Science:Math Exam Resources/Courses/MATH110/December 2014/Question 11
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Question 11 

Show that there is at least one point on the curve where the tangent line to the curve is horizontal. 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

The slope of the tangent line of the given curve at a point is . 
Hint 2 

Use the intermediate value theorem. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since the slope of the tangent line at a point is , to find the horizontal tangent line, it is enough to solve . Indeed, if we find a point satisfying , the tangent line of the given curve at is horizontal. First, we find the derivative using the power rule;
Now, the question is reduced to find a solution to . For this purpose, we use the intermediate value theorem; Observe that is a polynomial, and hence it is continuous for any point in the real line. On the other hand, by plugging some numbers, we get , . Then, by the intermediate value theorem, we can find a point such that . In other words, there exists a point such that , so that at that point, the corresponding tangent line is horizontal. 