Science:Math Exam Resources/Courses/MATH105/April 2013/Question 04 (a)
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Question 04 (a) 

Let . Find all critical points of . 
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Hint 

What is true about the partial derivatives of a multivariable function at critical points? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The critical points of a multivariable function are the values of where the following conditions are both simultaneously true: Since , the above conditions become: To solve these equations, we begin by factoring the lefthand sides, giving us Since the exponential function is never zero, the equations above are equivalent to We now proceed to determine the values of such both Eq. (1) and (2) hold. The solutions to Eq. (1) are and . We take each of these solutions and plug them into Eq. (2). If , then Eq. (2) becomes: Hence one of the critical points is . If , then Eq. (2) becomes: So, another critical point is . Therefore, the critical points of are 