Science:Math Exam Resources/Courses/MATH110/December 2016/Question 10
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Question 10 

Show that the tangent line to the curve at the point intersects the curve twice. 
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 tangent line of a function at the point is given by . 
Hint 2 

Two curves (including lines), say intersect if there are satisfying both equations. 
Hint 3 

In order to show that some equation have a solution, one can use the intermediate value theorem and check that such function is positive here and negative there. 
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 Hint 1, the tangent line to the curve at the point is Clearly, Also, since at we have This gives the equation of the tangent line, that is, We need to check that the tangent line intersects the curve twice. In other words, according to Hint 2, we need to find an such that (In such case, is automatically determined by either or .)
Therefore, if we write then it reamins to solve the equation We observe that This means that the tangent line intersects the curve at (and ).
By Hint 3, let us use the intermediate value theorem to find another intersection point. We need to find points where is positive and points where it is negative. Observe that Here, we have used the fact that , so that , that is, and .
Since is continuous everywhere, by intermediate value theorem, we conclude that for some . In summary, the tangent line intersect the curve at and at a point between and . Answer: 