query: How do I prove $\int_a^b |f(x)|^2 dx \leq \frac{(b-a)^2}{2} \int_a^b |f'(x)|^2 dx$.
positive_passages.docid: 22864
positive_passages.text: Show that $\int_{a}^{b} |f|^2\le\frac{(b-a)^2}{2}\int_{a}^{b} |f'|^2$
negative_passages.docid: 1626
negative_passages.text: How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$
negative_passages.docid: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 99753
negative_passages.text: How to prove $\int_0^1\frac{1-x}{(\ln x)(1+x)}\ dx=\ln\left(\frac2{\pi}\right)$?
negative_passages.docid: 62524
negative_passages.text: Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
negative_passages.docid: 149700
negative_passages.text: Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
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: 139111
negative_passages.text: Prove that $a<c<b$, $\frac{f(b)-f(a)-(b-a)f'(a)}{g(b)-g(a)-(b-a)g'(a)}=\frac{f"(c)}{g"(c)}$
negative_passages.docid: 98344
negative_passages.text: Prove that $\int_0^1[f''(x)]^2dx\ge4.$
negative_passages.docid: 44233
negative_passages.text: How to prove this inequality $ x + \frac{1}{x} \geq 2 $
negative_passages.docid: 136403
negative_passages.text: Prove that $\sqrt{a}\leq\frac{1+a}{2}$
negative_passages.docid: 97558
negative_passages.text: Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration
negative_passages.docid: 34081
negative_passages.text: Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$
negative_passages.docid: 156881
negative_passages.text: How prove this inequality $I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$
negative_passages.docid: 40700
negative_passages.text: Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$
negative_passages.docid: 15237
negative_passages.text: Prove an inequality using complex analysis
negative_passages.docid: 104394
negative_passages.text: What does $\int_{a}^{b} f(x)$ mean?