This excerpt comes from the textbook "Fundamentals of Differential Equations and Boundary Value Problems 7th Edition"
"The most transparent is due to Lagrange: if the (n+1)st derivative of f exists and is continuous on an interval containing $x_0$ and x, then $\epsilon_n(x)=\frac{f^{n+1}(\xi)(x-x_0)^{n+1}}{(n+1)!}$ where $\xi$ although unknown is guaranteed to lie between $x_0$ and $x$.
Figure 8.1 on page 422 and equation (6) suggest that one might control the error in the Taylor polynomial approximation by increasing the degree n of the polynomial (i.e., taking more terms), thereby increasing the factor (n + 1)! in the denominator. This possibility is limited, of course, by the number of times f can be differentiated. In Example 2, for instance, the solution did not have a fifth derivative at $x_0$ = 0 ($f^{(5)}(0)$ is “infinite”). Thus, we could not construct $p_5(x)$, nor could we conclude anything about the accuracy of $p_4(x)$ from the Lagrange formula."
Example 2: Initial value problem $$y''=3y'+x^2y; y(0)=10, y'(0)=5$$
The textbook then deduced that $y^{(5)} =3y^{(4)}+6y'+6xy''+x^2y'''$ and $y^{(5)}(0)=495$
So, my confusion arises because the excerpt stated that the solution did not have a fifth derivative but it does in this example. Is this a mistake on the textbooks part or am I misunderstanding?