query: For which positive numbers b is it true that $b^x \geq 1+x$ for all $x$?
positive_passages.docid: 26450
positive_passages.text: Why is $a=e$ the smallest number such that $a^x\ge 1+x$ for all $x$?
negative_passages.docid: 54211
negative_passages.text: If $A$ and $B$ are positive definite, then is $B^{-1} - A^{-1}$ positive semidefinite?
negative_passages.docid: 40420
negative_passages.text: Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$
negative_passages.docid: 139011
negative_passages.text: How to prove $B-A \succeq 0 \Leftrightarrow$ ellipsoid $x^TBx \leq 1$ contains $x^TAx \leq 1$?
negative_passages.docid: 108704
negative_passages.text: Help with proof of $|x+y| \geq |x|-|y|$
negative_passages.docid: 133496
negative_passages.text: is $\sqrt{x}$ always positive?
negative_passages.docid: 44799
negative_passages.text: Why is the even root of a number always positive?
negative_passages.docid: 52880
negative_passages.text: Intuition: If $a\leq b+\epsilon$ for all $\epsilon>0$ then $a\leq b$?
negative_passages.docid: 83386
negative_passages.text: Inequality with exponents $x^x+y^y \ge x^y +y^x$
negative_passages.docid: 66387
negative_passages.text: Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
negative_passages.docid: 70926
negative_passages.text: Showing that for $n\geq 3$ the inequality $(n+1)^n<n^{(n+1)}$ holds
negative_passages.docid: 139383
negative_passages.text: Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.
negative_passages.docid: 130327
negative_passages.text: Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
negative_passages.docid: 158844
negative_passages.text: If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
negative_passages.docid: 91617
negative_passages.text: Prove this inequality $a \sqrt{1-b^2}+b\sqrt{1-a^2}\le1 $
negative_passages.docid: 87567
negative_passages.text: How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$