query: There exists $c\in [a,b]$ such that $\int_a^c f(t)dt = \int_c^b f(t)dt$
positive_passages.docid: 18126
positive_passages.text: If $f \colon [a,b] \rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$
negative_passages.docid: 116389
negative_passages.text: If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute
negative_passages.docid: 69656
negative_passages.text: Prove or disprove that $\int_a^b |f(x)|\ \mathrm{d}x\geq \big | \int_a^b f(x)\ \mathrm{d}x\big| $
negative_passages.docid: 83798
negative_passages.text: Does $A$ a UFD imply that  $A[T]$ is also a UFD?
negative_passages.docid: 154612
negative_passages.text: $\int_{0}^{x} f(t) dt \ge {f(x)}$ holds or not on $[0,1]$
negative_passages.docid: 97184
negative_passages.text: Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $
negative_passages.docid: 141852
negative_passages.text: Prove that $2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$
negative_passages.docid: 61799
negative_passages.text: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
negative_passages.docid: 18261
negative_passages.text: Prove $\int_{a}^{b}{f(x)dx}>0$.
negative_passages.docid: 24810
negative_passages.text: Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
negative_passages.docid: 133266
negative_passages.text: If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis
negative_passages.docid: 90199
negative_passages.text: Prove that there exists a constant $c>0$ such that the following happens
negative_passages.docid: 143000
negative_passages.text: Show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$.
negative_passages.docid: 30694
negative_passages.text: Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$
negative_passages.docid: 15262
negative_passages.text: Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
negative_passages.docid: 34398
negative_passages.text: Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$
negative_passages.docid: 77667
negative_passages.text: A and B is similar ⇒ $A^T$ is similar to $B^T$.