query: For $x>0$, $x + \frac1x \ge 2$ and equality holds if and only if $x=1$
positive_passages.docid: 44233
positive_passages.text: How to prove this inequality $ x + \frac{1}{x} \geq 2 $
negative_passages.docid: 108782
negative_passages.text: Inequality $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1} \ge 1$
negative_passages.docid: 155341
negative_passages.text: Proving or disproving inequality $ \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y} \ge x + y + z $
negative_passages.docid: 36507
negative_passages.text: If $x,y>0$ and $x^2+y^3\ge x^3+y^4$, prove that $x^3+y^3 \le 2$.
negative_passages.docid: 27969
negative_passages.text: Why does $\frac{1}{x} < 4$ have two answers?
negative_passages.docid: 112565
negative_passages.text: Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$
negative_passages.docid: 82612
negative_passages.text: Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$
negative_passages.docid: 87055
negative_passages.text: How to prove $\frac{x}{1+x^2}0$?
negative_passages.docid: 52115
negative_passages.text: Show that $e^x > 1 + x + x^2/2! + \cdots + x^k/k!$ for $n \geq 0$, $x > 0$ by induction
negative_passages.docid: 59513
negative_passages.text: For $a, b \geq 0$, $0 < x < 1$, show $(a+b)^x \leq a^x + b^x$
negative_passages.docid: 35529
negative_passages.text: Prove that $\sin x+2x\ge\frac{3x(x+1)}{\pi}$ for all $x\in [0,\frac{\pi}{2}]$
negative_passages.docid: 77133
negative_passages.text: $g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that $g(xy) = g(x) + g(y)$ for all $x, y > 0$.
negative_passages.docid: 77026
negative_passages.text: Proof with inequalities
negative_passages.docid: 66247
negative_passages.text: Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$
negative_passages.docid: 152053
negative_passages.text: Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
negative_passages.docid: 156646
negative_passages.text: If $x$ is rational then $\,x+\frac{1}x \,$ is an integer $\iff x = \pm 1$
negative_passages.docid: 138632
negative_passages.text: Proof that if $a > 0$, then $\frac{1}{a} > 0$ using just field and order axioms