The variables u and v take positive values such that 1/u + 1/v = 1/20. If s = u + v, find the least value of s as u varies. Thanks
I don't really see how this is a differentiation problem.
$\displaystyle \frac{1}{u} + \frac{1}{v} = \frac{u + v}{uv}$.
So $\displaystyle \frac{u + v}{uv} = \frac{1}{20}$.
Therefore $\displaystyle u + v = 1$ and $\displaystyle uv = 20$.
Solving these equations simultaneously gives...
$\displaystyle u = 1 - v$
$\displaystyle (1 - v)v = 20$
$\displaystyle v - v^2 = 20$
$\displaystyle 0 = v^2 - v + 20$
This value does not have a solution... (check it's discriminant).
You are arguing that because a/b= c/d, "therefore" a= c and b= d. That is not true! 2/4= 1/2 but $\displaystyle 2\ne 1$ and $\displaystyle 4\ne 2$.
Solving these equations simultaneously gives...
$\displaystyle u = 1 - v$
$\displaystyle (1 - v)v = 20$
$\displaystyle v - v^2 = 20$
$\displaystyle 0 = v^2 - v + 20$
This value does not have a solution... (check it's discriminant).