Number 4
Directions: Find a positive number such that the sum of the number and its reciprocal is as small as possible.
My work:
Let x be the positive number
Let 1/x be its reciprocal.
Minimize x + 1/x
f ( x ) = x + 1/x
Because the derivative of 1/x =
derivative of \(\displaystyle x^{-1}\)
\(\displaystyle =(-1)x^{-2}\) \(\displaystyle =\frac{-1}{x^{2}}\)
f ( x ) = x + 1/x
f ‘ (x) = \(\displaystyle 1 + \frac{-1}{x^{2}}\)
f ‘ (x) =
0 = \(\displaystyle 1 + \frac{-1}{x^{2}}\)
-1 = \(\displaystyle \frac{-1}{x^{2}}\)
Multiply both sides by \(\displaystyle x^{2}\)
\(\displaystyle x^{2}\) * (-1) = \(\displaystyle \frac{-1}{x^{2}}\) * \(\displaystyle x^2\)
-\(\displaystyle x^2\)=-1
\(\displaystyle x^2\) = 1
x = \(\displaystyle \sqrt{1}\) (I am not sure if I am supposed to put the plus or minus sign here or just wait till the next step)
x = +1 or – 1
It has to be x = +1 because the directions say it is positive.
So the number is x=1
and the reciprocal 1/x = 1/1 = 1
That just seems like a really strange answer to me. That the number and the reciprocal are the same thing? That just seems like I got the wrong answer!
Directions: Find a positive number such that the sum of the number and its reciprocal is as small as possible.
My work:
Let x be the positive number
Let 1/x be its reciprocal.
Minimize x + 1/x
f ( x ) = x + 1/x
Because the derivative of 1/x =
derivative of \(\displaystyle x^{-1}\)
\(\displaystyle =(-1)x^{-2}\) \(\displaystyle =\frac{-1}{x^{2}}\)
f ( x ) = x + 1/x
f ‘ (x) = \(\displaystyle 1 + \frac{-1}{x^{2}}\)
f ‘ (x) =
0 = \(\displaystyle 1 + \frac{-1}{x^{2}}\)
-1 = \(\displaystyle \frac{-1}{x^{2}}\)
Multiply both sides by \(\displaystyle x^{2}\)
\(\displaystyle x^{2}\) * (-1) = \(\displaystyle \frac{-1}{x^{2}}\) * \(\displaystyle x^2\)
-\(\displaystyle x^2\)=-1
\(\displaystyle x^2\) = 1
x = \(\displaystyle \sqrt{1}\) (I am not sure if I am supposed to put the plus or minus sign here or just wait till the next step)
x = +1 or – 1
It has to be x = +1 because the directions say it is positive.
So the number is x=1
and the reciprocal 1/x = 1/1 = 1
That just seems like a really strange answer to me. That the number and the reciprocal are the same thing? That just seems like I got the wrong answer!