Lagrange Interpolating Polynomials - Error Bound

Let $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$. Construct interpolation polynomials of degree at most one and at most two to approximate $f(1.4)$, and find an error bound for the approximation.

So I know how to construct the interpolation polynomials, but I'm just not sure how to find the error bound. I know that the formula for the error bound is: $${f^{n+1}(\xi(x)) \over (n+1)!} \times (x-x_0)(x-x_1)...(x-x_n)$$

For the interpolation polynomial of degree one, the formula would be: $${f^{2}(\xi(x)) \over (2)!} \times (x-1)(x-1.25)$$

So if I take the second derivative of the function, I would get $f''(x) = 4e^{2x}$. Since $f''$ is strictly increasing on the interval $(1, 1.25)$, the maximum error of ${f^{2}(\xi(x)) \over (2)!}$ will be $4e^{2 \times 1.25}/2!$. Plugging in $x=1.4$ in the formula above gives us $1.461899$. I was just wondering if this is the correct way to calculate the error bound, since I've seen examples where they would take the derivative and find critical points and then I would get lost.

Can someone please clarify? Thanks!

I will do the part with all three points and you can do the other with two points.

We are given that $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$.

We are asked to construct the interpolation polynomial of degree at most two to approximate $f(1.4)$, and find an error bound for the approximation.

You stated that you know how to find the interpolating polynomial, so we get:

$$P_2(x) = 26.8534 x^2-42.2465 x+21.7821$$

The formula for the error bound is given by:

$$E_n(x) = {f^{n+1}(\xi(x)) \over (n+1)!} \times (x-x_0)(x-x_1)...(x-x_n)$$

Since we do not know where $\xi(x)$ is, we will find each error over the range and multiply those together, so we have:

$$\max_{(x, 1, 1.6)} |f'''(x)| = \max_{(x, 1, 1.6)} 8 e^{2 x} = 196.26$$

Next, we need to find:

$$\max_{(x, 1, 1.6)} |(x-1)(x-1.25)(x-1.6)| = 0.00754624$$

Thus we have an error bound of:

$$E_2(x) = \dfrac{196.26}{6} \times 0.00754624 \le 0.246838$$

If we compute the actual error, we have:

$$\mbox{Actual Error}~ = |f(1.4) - P_2(1.4)| = |15.0446 - 15.2698| = 0.2252$$

I know this is super late, but they use critical points to find the maximum of the absolute value of $(x−x_0)(x−x_1)\ldots(x−x_n)$ since we know that this product equals zero if $x= x_0, x_1,\ldots,\text{ or }x_n$.

This means that we don't consider the endpoints when finding the max in that interval, so the only possible choices are the critical points in that interval. And we know that there has to exist a critical point between each of the zeros so comparing the norms of each of the critical points always gives us the max for that part of the error.