Science:Math Exam Resources/Courses/MATH101/April 2018/Question 04 (ii)/Solution 1

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To find the positive points at which local minimums of occur, let's find the derivative of first.

Since can be written as a composite of two functions, , where , we can apply the chain rule to find its derivative. In this process, we need the derivative of .

By the Fundamental theorem of Calculus, we have

Then, the derivative of is

Since for any on the real line and we consider only positive , the sign of is determined by .

Note that at (i.e., ). Also,

Since the sign of is changed from minus to plus at for the first time among positive , so is the sign of . Therefore, the smallest number at which has a local minimum is .

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