Science:Math Exam Resources/Courses/MATH102/December 2011/Question 06
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Question 06 

Let and . Find all values of x where the tangent lines of these two functions have the same slope (or show that no such values exist). 
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 

Remember that tangent line slopes are just derivatives at points. So take derivatives and set them equal. 
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. First, we take derivatives and see that
Setting the derivatives equal to each other gives
which implies that
Hence, we have that or . The first gives the solution and the second gives both seen by exponentiating both sides. Both of these values are in the domain of the original function and so this completes the proof. 