Proving function is strictly positive in a interval defined by coefficients

Question

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}\dots+a_1x+1$. Prove that $f(x)$ is strictly positive if $$0<|x|<{1\over1+\sum_{i=1}^{n}|a_i|}.$$

Any hints on how to start?

Answer 1

Let $h(x)=f(x)-1=x\sum_{i=1}^{n}a_ix^{i-1}$ then for $|x| < \dfrac{1}{1 + \sum_{i=1}^n |a_i|}$we have that $|x|<1$ and $$|h(x)|\leq |x|\sum_{i=1}^{n}|a_i|<1.$$ Therefore $$f(x)=1+h(x)\geq 1-|h(x)|>0.$$