Science:Math Exam Resources/Courses/MATH104/December 2012/Question 01 (o)
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Question 01 (o) 

Show that the equation has a solution. 
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 

Apply the intermediate value theorem. 
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. Define the function as Notice that this function is continuous for all values of . Hence, we need only find a closed interval where the sign of changes and invoke the intermediate value theorem to get the desired conclusion. Consider and . If we take these two points and plug them into , we get and so the sign of changes over the closed interval . Therefore, there exists a value in such that by the Intermediate Value Theorem. 