Partial concave maximization of subset of variables

Question

Let $f(x_1, \dots, x_N)$ be a concave function in $x_1, \dots, x_N$. For arbitray $n>1$, prove that the (constrained) truncated function defined by $$g(x_1, \dots, x_{n-1}) = \max_{x_n, \dots, x_N \geq c} f(x_1, \dots, x_N),$$ where $c$ is a constant, is necessarily concave in $x_1, \dots, x_{n-1}$.