I want to prove, given $\gamma>0$ and $x\in \mathbb{R}^2_+$, if the utility function:
$$u(x)=x_1 x_2 + \gamma x_2$$
is concave, strictly concave, quasi-concave or strictly quasi-concave.
I have tried with Hessian but I do not reach anywhere.
Then I tried with the definition of concavity and quasi-concavity and I got that it is strictly quasi-concave if (supossing without loss of generality that $u(y)>u(x)$ ): $$\alpha(y_1-x_2)(y_2-x_2) < (y_1 y_2 + \gamma y_2)(x_1 x_2 + \gamma x_2)$$ Which I think is sattisfied.
Can someone give me a hint on this?