I was reading this MSE question, and I found the top answer (by the user Amzoti) very strange.
The question asks to interpolate $f(x) = e^{2x} - x$ at $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$, then approximate $f(1.4)$ using the polynomial, and finally calculate the error. The strange part is the error.
The well-known formula for the error is $$E(x) = \frac {1}{3!} f^{(3)}(\xi(x)) (x - 1)(x - 1.25)(x - 1.6).$$
What I think is: $$\begin{align*}|E(1.4)| &\le \frac 16 \left( \max_{y \in [1, 1.6]}|f^{(3)}(y)| \right)|(1.4 - 1)(1.4 - 1.25)(1.4 - 1.6)|\\ \\ & = \left( \max_{y \in [1, 1.6]}|f^{(3)}(y)| \right) (0.002). \end{align*} $$ However, Amzoti did
$$|E(1.4)| \le \frac 16 \max_{y \in [1, 1.6]} |f^{(3)}(y)| \cdot \max_{y \in [1, 1.6]} |(y - 1)(y - 1.25)(y - 1.6)|$$ This is
A) A worse error bound
B) More work
Isn't my approach correct? Am I missing something? Thank you very much.