query: $p(x)\geq 0 \forall x\Rightarrow p(x)+p'(x)+p''(x)+...+p^{(n)}(x)\geq 0$
positive_passages.docid: 1519
positive_passages.text: Sum of derivatives of a polynomial
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 126749
negative_passages.text: Evaluate $\lim_{x \rightarrow 0} x^x$
negative_passages.docid: 12781
negative_passages.text: Prove the following inequality $\sqrt[n]{x_1x_2...x_n}\ge (n-1)$
negative_passages.docid: 34349
negative_passages.text: Prove that $\frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n} \geq \frac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$
negative_passages.docid: 66387
negative_passages.text: Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
negative_passages.docid: 57639
negative_passages.text: Show there exists an $\alpha\in F_{p^n}$ such that $\alpha +\alpha^p+\alpha^{p^2}+\cdots+\alpha^{p^{n-1}}\neq 0$
negative_passages.docid: 139537
negative_passages.text: Show that $x\subseteq y \Leftrightarrow \mathcal{P}(x)\subseteq\mathcal{P}(y)$.
negative_passages.docid: 154945
negative_passages.text: $|px^2+qx+r|\le1$ for all $x$ in $[-1; 1]$, show that $|rx^2+qx+p|\le2$ for all $x$ in $[-1; 1]$
negative_passages.docid: 30386
negative_passages.text: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
negative_passages.docid: 34907
negative_passages.text: Show that $\int_{0}^{1}\{P(x)\}^{2}\,dx = (n + 1)^{2}\left(\int_{0}^{1}P(x)\,dx\right)^{2}$
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 66996
negative_passages.text: Proof by induction of Bernoulli's inequality $ (1+x)^n \ge 1+nx$
negative_passages.docid: 122115
negative_passages.text: Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
negative_passages.docid: 88687
negative_passages.text: Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
negative_passages.docid: 25187
negative_passages.text: Why $∀x(P(x)→Q)$ is equivalent to $∃xP(x)→Q$