Is $f$ concave at first argument

Question

Is $f:\mathbb{R}^n\times\mathbb{R}^n\longrightarrow\mathbb{R}$ with $f(x,y)=\sum\limits_{j=1}^nx_j^3(y_j-x_j)$ concave in its first argument?

Answer 1

Consider $n=1$. Restrict $f(x,y)=x^3y-x^4$ to the line $y=c$. Compute $$ (x^3c-x^4)''= 6cx - 12x^2 \tag1$$ If $c\ne 0$, the right hand side of (1) changes sign at $0$.