Asked by Anonymous
For x>0, what value of x gives the least sum of x and its reciprocal? Show work.
Answers
Answered by
michael
x+(1/x) = y
y&prime = 1 - 1/x<sup>2</sup>
set that equal to zero.
You didn't specify that you were in calculus though, so I'll provide a second way.
x + (1/x) = y
if x = 1, then y = 2
if x = 1/2 then y = 5/4>2
if x = 2 then y = 5/4>2
if x = 1/3 then y = 10/3>2
if x = 3 then y = 10/3>2
you can see that as x gets bigger and smaller then 1 y gets bigger then 2, so when x = 1 you have a minimum value
y&prime = 1 - 1/x<sup>2</sup>
set that equal to zero.
You didn't specify that you were in calculus though, so I'll provide a second way.
x + (1/x) = y
if x = 1, then y = 2
if x = 1/2 then y = 5/4>2
if x = 2 then y = 5/4>2
if x = 1/3 then y = 10/3>2
if x = 3 then y = 10/3>2
you can see that as x gets bigger and smaller then 1 y gets bigger then 2, so when x = 1 you have a minimum value