Let $f$ be a $C^2$ function such that $f(x),f'(x),f''(x)>0$ on $[1, \infty)$. Then I want to show that $$\sum_{k=1}^{n} f(k)-\int_{1}^{n}f(x)\ dx- \frac{f(n)}{2} - \frac{f(1)}{2} \leq \frac{f'(n)}{4}$$ for all natural numbers $n$. However, I cannot get any motivation of how the term $\frac{f'(n)}{4}$ appears in this estimate...Could anyone please help me?
2026-03-27 22:01:31.1774648891
Geometric estimate of a function
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