A Tangent Dominates the Curve

Let $f(x)=nx-x^n$ for $x\in\mathbb R$, where $n\in\mathbb N^{*}$ and $n\geqslant 2$.

1. Discuss the monotonicity of $f(x)$. 2. Let $P$ be the intersection point of the curve $y=f(x)$ with the positive $x$-axis, and let $y=g(x)$ be the tangent line to the curve at $P$. Prove that $f(x)\leqslant g(x)$ for every positive real $x$. 3. Suppose the equation $f(x)=a$ (with $a$ real) has two positive real roots $x_1,x_2$. Prove that