query: Prove the inequality $f'(0) \ge -\sqrt {2}$
positive_passages.docid: 149764
positive_passages.text: If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$
negative_passages.docid: 58969
negative_passages.text: A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$
negative_passages.docid: 82567
negative_passages.text: Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
negative_passages.docid: 103030
negative_passages.text: Proving $\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}$ for $x> 0$
negative_passages.docid: 117262
negative_passages.text: Proving a logarithm inequality
negative_passages.docid: 15232
negative_passages.text: Proving an inequality involving real powers
negative_passages.docid: 98344
negative_passages.text: Prove that $\int_0^1[f''(x)]^2dx\ge4.$
negative_passages.docid: 104263
negative_passages.text: How to show that $f'(x)<2f(x)$
negative_passages.docid: 18016
negative_passages.text: Proof of the following inequality $ \frac{x - y}{\log x - \log y} > \sqrt{xy} $, $x>y$.
negative_passages.docid: 20192
negative_passages.text: Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
negative_passages.docid: 128495
negative_passages.text: Integral Inequality $|f''(x)/f(x)|$
negative_passages.docid: 27570
negative_passages.text: Prove $(1-x)\ln(1-x)+(1+x)\ln(1+x)\leq 2x^2$ for $0<x<1$
negative_passages.docid: 8207
negative_passages.text: Can I prove this inequality algebraically?
negative_passages.docid: 90896
negative_passages.text: proof of inequality $e^x\le x+e^{x^2}$
negative_passages.docid: 63526
negative_passages.text: Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
negative_passages.docid: 145496
negative_passages.text: Show $\lim_{h \to \ 0} \frac{f(x + 2h) - 2f(x+h) + f(x)}{h^{2}} = f''(x)$ Proof verification
negative_passages.docid: 42976
negative_passages.text: How do you prove the follwing $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$