Question

find a positive number x such that the sum of 16x and 1/x is as small as possible using optimization

Answer 1

\[f(x)=16x+\frac{1}{x}\]and you want the min right? for \(x\geq 0\)

Answer 2

then
\[f'(x)=16-\frac{1}{x^2}\] find the critical points via
\[16-\frac{1}{x^2}=0\]
\[\frac{1}{x^2}=16\]
\[16x^2=1\]
\[x^2=\frac{1}{16}\]
\[x=\frac{1}{4}\]

Answer 3

kind of a long winded way of solving but it is late, you can probably do it shorter. notice i ignored the negative root because you are only concerned with \(x>0\)

Answer 4

thanks so much