How do you find a positive number such that the sum of the number and its reciprocal is as small as possible?

Answer 1

The smallest sum of a number #n# and its reciprocal #1/n# is #2# which occurs when #n = 1#. Any other value of #n# will produce a larger sum.

Let us consider a positive number #n#, making sure #n \ne 0# so that we don't have an undefined reciprocal.
We want to find a #1/n# such that #n + 1/n# is minimized. We can call this sum a function #f(n) = n + 1/n#.
Now we take the derivative of #f(n)# w.r.t. #n# and set it equal to zero to obtain the minimum.
#f'(n) = 1 -1/n^2#
#1 - 1/n^2 = 0# #1 = 1/n^2# #n^2 = 1# #n = +- 1#
However, we reject the negative value as #n > 0#. Hence, #n = 1#.
So the minimum sum obtainable is #f(1) = 1+ 1/1 = 2#
Hence, the smallest sum of a number #n# and its reciprocal #1/n# is 2 when #n = 1#. Any other value of #n# will produce a larger sum.
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Answer 2

To find the positive number ( x ) such that the sum of the number and its reciprocal is as small as possible, you need to minimize the expression ( x + \frac{1}{x} ). This is done by finding the critical points of the function and determining whether they correspond to a minimum. To do this, you can take the derivative of the expression with respect to ( x ), set it equal to zero, and solve for ( x ). After finding the critical points, you can use the second derivative test to determine whether they correspond to a minimum or maximum. Alternatively, you can complete the square to find the minimum value. The result will be ( x = 1 ), which corresponds to the minimum sum of the number and its reciprocal.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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