Have I calculated the error bound incorrectly?
Question
Use the Trapezoidal Rule error, to find the smallest reasonable integer $n$ such that $E_T \leq \frac{1}{10}$ of $$\int_{1}^{3} 2\ln(t)dt$$
My work:
\begin{align}f(x) &= 2\ln(t)\\ f’(x) &= \frac{2}{x}\\ f''(x) &= - \frac{2}{x^{2}}\end{align}
Testing the end points, should I find critical points? Not sure
\begin{align}f''(1)& = \frac{2}{9}\\\\ f''(3) &= 2 \tag{larger value therefore max}\\\\ |E_T| &< \frac{(2)(2)^{3}}{12n^{2}} < \frac{1}{10}\\\\ \frac{16}{12n^{2}} &< \frac{1}{10}\\\\ \sqrt{\frac{40}{3}} &< n\end{align}
Is this incorrect? Thank you