Weierstrass Approximation Theorem with an additional condition

Question

Good afternoon:

Let $f$ be a continuous function defined on an closed interval $[a, b]\subset\mathbb R$. By Weierstrass Approximation Theorem, for any $\epsilon>0$, there is a polynomial $p$ such that $|p(x)-f(x)|<\epsilon$ for all $x\in [a, b]$.

My question is: if in addition we assume that $f(b)>0$ and $f(b)>f(x)$ for $x\in [a, b)$, can we choose the polynomial $p$ with the additional reqirement that $p(x)>\frac{f(b)}{2}$ for $x\in [b, +\infty)$?

My area is Several Complex Varables and the quetion is motivatied by the study of holomorphic transformations of certain bounded domains.

Thank you very much for your help!

Answer 1

Sure. Assume by simplicity $a=0, b=1$. If $p$ is any uniform approximation of $f$ (say with $p(b) \ge f(b) )$ just add $x^N/N$ to $p$, with $N$ large enough.