Science:Math Exam Resources/Courses/MATH105/April 2012/Question 07 (a)
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Question 07 (a) 

Consider (a) Find all critical points of this function. 
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 

Both partial derivatives, ƒ_{x} and ƒ_{y} must vanish (equal 0) at a critical point. 
Hint 2 

Remember to use the product rule. Also, e^{y} does not vanish for any value of y. 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. To find the critical points of , we need to set both partial derivatives to 0: and, using the product rule, To solve the simultaneous system, we should first factor the equations as best we can (this makes it easier to find conditions that are necessary for the derivatives to vanish). Nothing can be done for the first equation so we leave it: For the second equation, we write: It's easier to work with the equation above first. From above, we consider the possibilities that:
Let's now use the possibilities above in the equation (1): ƒ_{x = yey + x = 0. }
