Science:Math Exam Resources/Courses/MATH100/December 2019/Question 12
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Question 12 

How many solutions does the equation have? Justify your answer. 
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 zeros of the function . Use the intermediate value theorem to find its zeros. 
Checking a solution serves two purposes: helping you if, after having used the hint, 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. Solutions of the equation are zeros of the function . The domain of is . Since , is strictly decreasing when and strictly increasing when . This implies that has a local minimum at , and . (This is because , so .) Finally, we have that and . By the intermediate value theorem, has at least one zero in the interval ; and at least one zero in the interval . But is strictly decreasing on ; and strictly increasing on . Therefore the function has exactly two zeros: one in the interval and another one in the interval . 