Science:Math Exam Resources/Courses/MATH100/December 2015/Question 12 (b)
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Question 12 (b) |
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Consider the equation . Show that this has at least two solutions. |
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 |
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Use 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.
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Solution |
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Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Recall the intermediate value theorem: suppose that is continuous on and let be any number between and , where . Then, there exists a number such that .
Observe that , , . Hence we choose the intervals and because changes sign at the endpoints of each.
Similarly, since is between and , by the intermediate value theorem, there exists such that . Therefore, there exists at least two solutions to the given equation. |