I am struggling a little to figure out the best way to find the concavity and inflection points of the below function (and other similar functions)
$f(x)=-\ln(5x^2 + 6)$
First I get $f'(x)=\frac{-10}{5x^2+6}$
Then $f''(x)=\frac{10(5x^2+6)}{(5x^2+6)^2}$
From here I get a little lost on how to find the interval for $f''(x)$ in order to find where $f''(x)$ is concave up or down and the inflection points. I have been setting it to zero but not been able to solve.
I am wondering if someone can help guide me through the best way to solve these types of problems? Or if you have a good recommendation on study material?
Best Answer
I have got $$f''(x)=\frac{10 \left(5 x^2-6\right)}{\left(5 x^2+6\right)^2}$$ and with $$f''(x)=0$$ you will get the inflection points $$x_w$$.(If $$f'''(x_w)\neq 0$$