pxchg1200

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I love mathematics very much! especially analysis,and inequality.Here is my blog,Welcome to visit. :D

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12 Questions

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15	Prove $f(x)=ax+b$
13	Show $ \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{1}+a_{2}+\cdots+a_{n}}}=1 $ if $\displaystyle \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{n}}}=1 $
10	Prove that $f(x)$ be a constant
10	Evaluate $ \int_{0}^{1}{\frac{\ln{x}}{x^2-x-1}dx}$
8	Integrate $\displaystyle \int_{0}^{\pi}{\frac{x\cos{x}}{1+\sin^{2}{x}}dx}$

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446 Reputation

+15	Prove $a\sqrt[3]{a+b}+b\sqrt[3]{b+c}+c\sqrt[3]{c+a} \ge 3 \sqrt[3]2$
-2	Prove $f(x)=ax+b$
+5	Integrate $\int_{0}^{1}{x^{-x}(1-x)^{x-1}\sin{\pi x}dx}$
+5	Prove that $f(x)$ be a constant

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5 Answers

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2	Prove $a\sqrt[3]{a+b}+b\sqrt[3]{b+c}+c\sqrt[3]{c+a} \ge 3 \sqrt[3]2$
1	Prove that $\int_{0}^{1}{f^{2}(x)dx}\leq \frac{4}{3}\left(\int_{0}^{1}{f(x)dx}\right)^2$
1	Showing that: $(\frac{a}{b+c})^2+(\frac{b}{a+c})^2+(\frac{c}{a+b})^2+\frac{10abc}{(a+b)(b+c)(c+a)}\ge 2$
0	proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.
0	Prove that $\frac 1 {x+y}+\frac 1 {y+z}+\frac 1 {z+x}\geq \frac 5 2$.

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3 inequality × 4	0 integration
1 calculus × 13	0 limit
1 homework	0 integral
0 definite-integral	0 polynomials
0 real-analysis

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