I attempted a solution using this version of Taylor's Theorem, \begin{align*} |f(d) - P_n(d)| &= \left| \frac{f^{n+1}(t)}{(n+1)!} \right| |d-c|^{n+1} \\ |f(d) - f(c) - f'(c)(d-c)| &= \left| \frac{f''(t)}{2} \right| |d-c|^2 \end{align*} Letting $d=1$ and $c=1/2$ and using the triangle inequality, I get $$ \frac{|f''(t)|}{4} - \frac{|f(d)-f(c)|}{|d-c|} \geq -|f'(c)| $$
The negative sign on the RHS is throwing me off because otherwise I could easily achieve the desired result. Am I attempting using the wrong theorem or are there any glaring mistakes in my process?