Separating Lines, Three Ways

Functions $y=f(x)$, $y=g(x)$, and $h(x)=kx+b$ ($k,b\in\mathbb R$) satisfy

1. If $f(x)=x^2+2x$, $g(x)=-x^2+2x$, and $D=(-\infty,+\infty)$, find $h(x)$. 2. If $f(x)=x^2-x+1$, $g(x)=k\ln x$, $h(x)=kx-k$, and $D=(0,+\infty)$, find the range of $k$. 3. If $f(x)=x^4-2x^2$, $g(x)=4x^2-8$, $h(x)=4(t^3-t)x-3t^4+2t^2$ (where $0<|t|\leqslant\sqrt 2$), and $D=[m,n]\subseteq\left[-\sqrt 2,\sqrt 2\right]$, prove that $n-m\leqslant\sqrt 7$.