This is a homework question I was asked to do
Of a twice differentiable function $ f : \mathbb{R} \to \mathbb{R} $ it is given that f(2) = 3, f'(2) = 1 and f''(x) = \frac{e^{-x}}{x^2+1} . Now I have to prove that $ \frac{7}{2} \leq f\left(\frac{5}{2}\right) \leq \frac{7}{2} + \frac{e^{-2}}{40} . $ I tried this by computing the third Taylor polynomial of $f$ near $a=2$, setting $x = \frac{5}{2}$, which gave me $f(5/2) \approx 7/2 + \frac{e^{-2}}{40} - \frac{ - e^{-5/2}}{48} $, but now I don't know what to do next. I guess one has to do something with finding the error of the first and second order Taylor polynomials, but I'm not sure how to do so. Can you help me?
Thanks in advance,