Need help on answering proof question on polynomials p1,p2 of degree

Question

I am not sure how to prove this and would appreciate some help on how to proof this statement.

Assume that $f : [0, \infty)\rightarrow\mathbb{R}$ is continuous and differentiable, and $f′(x)$ is bounded with :

$\sup {|f′(x)|} = M < \infty$.

Show that there exist polynomials $p_1$,$p_2$ of degree one such that $p_1(x) \leq f(x) \leq p_2(x)$ for all $x \in [0,\infty)$.

Thanks!

Answer 1

Use mean value theorem to see that for all $x > 0$,

$$\underbrace{f(0) - M x}_{=: p_1(x)} \leqslant f(x) \leqslant \underbrace{f(0) + M x}_{=: p_2(x)} $$

Answer 2

Now, since $|f'(x)|<M$ for all $x\in\mathbb{R}$, define \begin{align*} g(x)&=f'(x), \quad f'(x)\leq 0\\ &=0, \quad f'(x)> 0 \end{align*} and \begin{align*} h(x)&=f'(x), \quad f'(x)> 0\\ &=0, \quad f'(x)\leq 0 \end{align*} Integrate these functions separately, and with the bounded inequality in mind,i.e.,

$$G(x)=\int g(x)dx\geq-\int Mdx \quad \quad H(x)=\int h(x)dx<\int Mdx$$

Then, see that $$f(x)=G(x) + H(x),$$ and so, $$c-Mx\leq f(x)\leq c+Mx$$