Question

Find two positive numbers that satisfy the given requirements.
The second number is the reciprocal of the first and the sum is a minimum.

Answer 1

minimize
\[x+\frac{1}{x}\]

Answer 2

take the derivative and find the critical points

Answer 3

\[y=x+\frac{1}{x}\]
\[y'=1-\frac{1}{x^2}\] set
\[1-\frac{1}{x^2}=0\] get
\[1=\frac{1}{x^2}\] so
\[x=\pm1\] by inspection. since you want a positive number the answer is 1

Answer 4

which frankly you could have arrived at by ignoring calc and just thinking