Consider the following function $f: \mathbb{R^3} \rightarrow \mathbb{R}$ defined by
$f(x_1, x_2, x_3) = 2x_1^2 +5x_2^2 +3x^3_3 -\ln(|x_3| +1) + \left(\frac{x_1+x_2}{x_3^2+1} \right )^2$
Determine if $f$ is convex or not.
I suspect that the function is indeed convex, for this I have tried to find the Hessian matrix and see that it is positive semidefinite. However the function is quite complex for that calculation, even using the Sylvester criterion. In any case, I cannot conclude that the function is convex. Could you help me?