find estimation of interpolation error for non differential function

Question

Given $f(x)=|x|^{1/2}$ , $-1\le x\le 1$ , I have found the interpolating polynomial $ p(x)=x^2$ for $x_{0}=-1,x_{1}=0,x_{2}=1$. How to estimate $$\max_{-1\le x\le 1}|f(x)-p(x)|$$ now that $f$ is not $C^3$?

Answer 1

Set $t=\sqrt{|x|}$ and use the symmetry so that you now have to determine $$ \max_{t\in[0,1]}|t-t^4|. $$

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Note that $a+bx^2$ for some $a>0$, $b<1$ might result in a better approximation, i.e., a smaller max-norm, if no interpolation is required.