Find the point on the curve y=sqrt(1/2ln(1/x)) that is closest to the origin
I have attempted to do this question a couple of times, but I can't get the right answer. Please help.
Thanks in advance.
$\displaystyle D^2 = (x - 0)^2 + (y - 0)^2$
let $\displaystyle D^2 = u$ ... minimizing $\displaystyle u$ will minimize $\displaystyle D^2$
$\displaystyle u = x^2 + \frac{1}{2} \ln\left(\frac{1}{x}\right)$
$\displaystyle u = x^2 - \frac{1}{2} \ln{x}$
$\displaystyle \frac{du}{dx} = 2x - \frac{1}{2x} = 0$
$\displaystyle 4x^2 - 1 = 0$
$\displaystyle x = \frac{1}{2}$
$\displaystyle \frac{d^2u}{dx^2} = 2 + \frac{1}{2x^2} > 0 \implies$ critical value is a minimum
I'll leave you to find the y-coordinate of the point.