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For the function $f(x) = \mathrm{e}^x$ on the interval $[0,1]$, by using polynomial interpolation with $x_0 = 0$, $x_1 = 1/2$, and $x_2 = 1$, find the upper bound for the magnitude $$ \max_{0 \leq x \leq 1} |\mathrm{e}^x - p_2(x)| $$

I'm confused on how to solve this.

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If you have got the expression of $p_2(x)$, then finding the maximum of $|e^x-p_2(x)|$ for $x\in[0,1]$ can be converted to the following question:$$\max_{x\in[0,1]} (e^x-p_2(x))$$ or $$\min_{x\in[0,1]} (e^x-p_2(x))$$ Then just let $g(x)=e^x-p_2(x)$ and find the critical points in $[0,1]$.

Hope this can help you.

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