query: $f(b)-f(a) =((b-a)/2)\cdot(f'(a)+f'(b))-((b-a)³/12)\cdot f'''(c)$
positive_passages.docid: 150057
positive_passages.text: Approximating a three-times differentiable function by a linear combination of derivatives
negative_passages.docid: 133682
negative_passages.text: Prove that if $f(t)$ satisfies $f(t)=f'(t)$, then $f(t+s)=f(t)f(s)$ for all $t,s\in R$ without using the fact that $f(t)=c\cdot e^t$.
negative_passages.docid: 80421
negative_passages.text: Show that $a - b \mid f(a) - f(b)$
negative_passages.docid: 111186
negative_passages.text: Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
negative_passages.docid: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 63750
negative_passages.text: let $f(\frac{x}{3})+f(\frac{2}{x})=\frac{4}{x^2}+\frac{x^2}{9}-2$ then find $f(x)$
negative_passages.docid: 53024
negative_passages.text: $f(x)f(1/x)=f(x)+f(1/x)$
negative_passages.docid: 129480
negative_passages.text: Prove that inf(A+B) = infA + infB
negative_passages.docid: 139019
negative_passages.text: Polynomial satisfying $f(x) = f'(x) \cdot f''(x)$
negative_passages.docid: 103706
negative_passages.text: If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
negative_passages.docid: 34077
negative_passages.text: Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f\big(x-f(y)\big)=f\big(f(y)\big)+x\cdot f(y)+f(x)-1$ for all $x,y\in\mathbb{R}$.
negative_passages.docid: 98933
negative_passages.text: Calculating the sum of $f(x) = \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^n}$
negative_passages.docid: 50655
negative_passages.text: Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$
negative_passages.docid: 65923
negative_passages.text: How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
negative_passages.docid: 82024
negative_passages.text: If $f(x)\cdot f(f(x)) = 1$ for all $x \in \mathbb{R},$ and if $f(10) = 9$ then find the value of $f(5)$
negative_passages.docid: 81794
negative_passages.text: Prove that $(ab,cd)=(a,c)(b,d)\left(\frac{a}{(a,c)},\frac{d}{(b,d)}\right)\left(\frac{c}{(a,c)},\frac{b}{(b,d)}\right)$
negative_passages.docid: 104263
negative_passages.text: How to show that $f'(x)<2f(x)$