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 .
Answer: