Science:Math Exam Resources/Courses/MATH110/December 2017/Question 09
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Question 09 

Use the Intermediate Value Theorem to show that there is a number in the interval such that the tangent line to the graph of at is parallel to the tangent line to the graph of at . Is closer to 0 or 2? 
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 

Recall that the Intermediate Value Theorem states that any value, between the functional values on the two ends of an interval, is attained by the function at some point inside the interval. 
Hint 2 

For two tangent lines to be parallel, the slopes (that is, the derivatives of the functions) must be equal. 
Hint 3 

We need to find a solution to the equation in the interval . 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. By the Hints, we need to solve the equation on the interval . First we compute So we need to find a root of the equation or Let . In order to apply the Intermediate Value Theorem, we want to check that is between and . Indeed, Therefore . Since is continuous, by the Intermediate Value Theorem, there exists a lying on the interval such that . In other words, there is a point at which and have parallel tangent lines.
there exists a point on at which vanishes. We can consider this point as . Then, is closer to .
