Use mean value theorem to prove $(1+x)^3 \ge (1+2x)$, where $x\ge 1$ We know $(1+x)^3 \ge (1+3x) \ge (1+2x)$.
Also we take $f(x)=(1+x)^3$, we can prove $f'(x)=3(1+x)^2\ge (1+2x)$ But I don't have any idea how to use MVT for this. Can anyone help?

Note that $f'(x)\geqslant12\geqslant2$ if $x\geqslant1$. So, if $x\geqslant1$, then, by the mean value theorem,$$f(x)-f(1)\geqslant2(x-1)(\iff f(x)\geqslant2x-2+8\geqslant2x+1).$$
