I'm analysing function $$y=x+\frac{\ln x}{x}$$
and I kinda don't get few parts. I get an oblique asymptote $y=x$, that function is increasing throughout the domain and that it is convex on $(e^\frac{3}{2}, \infty)$ and concave on $(0, e^\frac{3}{2})$
what bugs me is when i try to sketch a graph. It doesn't seem to match one on wolframalpha, particularly, it seems their is not convex on any part. One more thing, it also doesn't say function has an oblique asymptote.
Can anyone tell me how is wolfram getting those results?
Your calculation is correct, and what is more, Wolfram alpha is also correct. The reason why you don't see the graph being convex is that it approaches its asymptote ($y=x$) rather quickly, so it is hardly distinguishable from straight line by just graph inspection. But at least that verifies that your asymptote is correct, you can draw the $y=x$ next to it for comparison (something like
plot x+log(x)/x and x from x = 0 to 10).Personally I would suggest to be careful when using Wolfram or other CAS systems and only use it to check your work, not to rely on it too much.