I'm trying to sketch the function $$f(x)=\frac{\ln(5x^2+x)}{2-3x}$$ .
When I do the first derivative I get $$\frac{d}{dx}\bigg(\frac{\ln(5 x^2 + x)}{2 - 3 x}\bigg) = \frac{10 x + 1}{(2 - 3 x) (5 x^2 + x)}+\frac{3\ln(5 x^2 + x)}{(2 - 3 x)^2}$$ .
Then I wanted to solve the inequality $ f'(x)>0 $ (where $f'(x)$ is the first derivative) to see if there is any max/min point, and then i would like to solve $ f''(x)>0 $ to see if the concavity is convex or concave.
The problem is i couldn't solve both $ f'(x)>0 $ and $ f''(x)>0 $ .
It's 3 days I'm looking for a way to solve it and I understood that it is a Transcendental Function and that there are some way to approximate a solution with e.g. Newton's Method (which is pretty cool), but it is a bit convoluted since i should also find the third derivative in order to solve the second derivative inequality.
My question is: Is there an easy way to find max, min and concavity of a function where its first and second derivative are Transcendental functions?
Thanks for any hint