$\displaystyle F(x)=\frac{ln(x)}{8\sqrt{x}}$

We're supposed to find:

~The intervals on which f is increasing and decreasing

~The local maximum of f

~The inflection point

~The interval on which f is concave up

~The interval on which f is concave down

For the first one, I know that we're supposed to use the critical points to "divide" the function into invervals, but I don't understand it in terms of this function.

Even if you graph it, anything x <= 0 doesn't work, and if you take the derivative to find the critical points:

$\displaystyle F'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

f(x) = ln x

f'(x) = $\displaystyle \frac{1}{x}$

g(x) = $\displaystyle 8\sqrt{x}$

g'(x) = 4x^(-1/2) = $\displaystyle \frac{4}{\sqrt{x}}$

$\displaystyle F'(x) = \frac{\frac{8\sqrt{x}}{x} - \frac{4lnx}{\sqrt{x}}}{64x}$

Which I think raises a similar problem, because F'(x) is 0 at x = 0 and undefined x <= 0 because of the ln. So what am I supposed to do? I'm hoping that the rest of the problem is doable once the first bullet is taken care of, but I don't know.