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Bất đẳng thức và cực trị

Hệ thống nhắc nhở thành viên

Posted by Nesbit

Đặt tiêu đề thế nào để bài không bị xóa?

Posted by E. Galois

Trình soạn thảo công thức Toán

Posted by Nesbit

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Bất đẳng thức và cực trị

Hệ thống nhắc nhở thành viên

Đặt tiêu đề thế nào để bài không bị xóa?

Trình soạn thảo công thức Toán

$\frac{\sum a^2}{\sum ab}+\frac{1}{3}\geq\frac{8}{9}\sum\frac{a}{b+c}$

$T = (a-1)^3+(b-1)^3+(c-1)^3$

Chứng minh rằng:$8(2-a-b)(2-b-c)(2-c-a)\geq a^2b^2c^2$

$\sum\sqrt{\frac{a}{ab+3}}\leq\frac{3}{2}.$

$\frac{19}{20} \leq \sum \frac{1}{1+a+b^2} \leq \frac{27}{20}$

Tìm GTLN $$P=\frac{(2x^2+y+2023)^3}{2024(x^2+y^2)-2024(x+y)+3036}$$

Min $\sum\frac{1}{a^2+1}$

$\sum (a+b)^{4}\geq \frac{4}{7}\sum a^{4}$

$\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{x+2z}{x+z} \geq \frac{5}{3}$

$ T = abc.$

Tìm $GTNN$ của $\left | 5x^{2}+11xy-5y^{2} \right |$

$$\sqrt[n]{ka+m}+\sqrt[n]{kb+m}+\sqrt[n]{kc+m}$$

Tìm $GTLN$ của: $A=a^2+b^2+c^2-4a-4b-c.$

$\sum\frac{a}{b+c^2}\geq\frac{3}{2}$

Cho $a^2\geq 2b> 0$. Tìm GTNN của $A=\frac{a^6+4b^3}{a^2b^2}$

$GTLN$ $A=\frac{11b^3-a^3}{ab+4b^2}+\frac{11c^3-b^3}{bc+4c^2}+\frac{11a^3-c^3}{ca+4a^2}$

$2({a^{m+n}+b^{m+n}})\ge (a^m+b^m)(a^n+b^n)$

$\sum \frac{a_1{}}{({a_1+{a_2+...+a_n{}{}}{}})-{a_1{}}}\geq \frac{n}{n-1}$

Tìm $GTLN$ của tích $3$ số.

$P=\sqrt{a+b^2c^2}+\sqrt{b+c^2a^2}+\sqrt{c+a^2b^2}$

$\frac{x^{3}+y^{3}+z^{3}}{3}\geq xyz+\frac{3}{4}|(x-y)(y-z)(z-x)|$

$4\sum \frac{a}{a+b}+\frac{\sum ab^2 + abc}{\sum a^2b + abc} \geq 7$

$(a-b)^4+(b-c)^4+(c-a)^4\geq \frac{81}{128}c^4$

$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\geq \frac{3}{2}.\frac{a^3+b^3+c^3}{a^2+b^2+c^2}$

Tìm giá trị nhỏ nhât của $A=\frac{2x^2-2xy+9y^2}{x^2+2xy+5y^2}$

$\frac{2c+4a+15}{3b+1}+\frac{4a+3b+15}{2c+1}+\frac{3b+2c+15}{4a+1}\geq 15$

CMR $\sum \frac{1}{x+yz}\leq \frac{9}{2(xy+yz+zx)}$

$\frac{2xy+3}{x^2 +xy + y^2} + \frac{2yz+3}{y^2 +yz + z^2}+ \frac{2zx+3}{z^2 +zx + x^2} \geq 5 $

$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+3 \ge 2(a+b+c)$

min $P=\sum \frac{a^{2}b^{2}}{c(a^{2}+b^{2})}$

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Topic	Stats	Last Post Info
$\frac{\sum a^2}{\sum ab}+\frac{1}{3}\geq\frac{8}{9}\sum\frac{a}{b+c}$ Started by habcy12345, 17-03-2024	3 replies 1814 Views	$$\frac{\sum a^2}{\sum ab}+\frac{1}{3}\geq\frac{8}{9}\sum\frac{a}{b+c}$ - last post by habcy12345$ habcy12345 18-03-2024
$T = (a-1)^3+(b-1)^3+(c-1)^3$ Started by Giabao209, 17-03-2024	1 reply 752 Views	Duc3290 17-03-2024
Chứng minh rằng:$8(2-a-b)(2-b-c)(2-c-a)\geq a^2b^2c^2$ Started by MHN, 15-03-2024	2 replies 764 Views	$Chứng minh rằng:$8(2-a-b)(2-b-c)(2-c-a)\geq a^2b^2c^2$ - last post by dinhvu$ dinhvu 15-03-2024
$\sum\sqrt{\frac{a}{ab+3}}\leq\frac{3}{2}.$ Started by nguyenhuybao06, 13-03-2024	0 replies 769 Views	$$\sum\sqrt{\frac{a}{ab+3}}\leq\frac{3}{2}.$ - last post by nguyenhuybao06$ nguyenhuybao06 13-03-2024
$\frac{19}{20} \leq \sum \frac{1}{1+a+b^2} \leq \frac{27}{20}$ Started by Duc3290, 12-03-2024 bất đẳng thức, hoán vị	0 replies 753 Views	$$\frac{19}{20} \leq \sum \frac{1}{1+a+b^2} \leq \frac{27}{20}$ - last post by Duc3290$ Duc3290 12-03-2024
Tìm GTLN $$P=\frac{(2x^2+y+2023)^3}{2024(x^2+y^2)-2024(x+y)+3036}$$ Started by MHN, 11-03-2024	1 reply 1132 Views	$Tìm GTLN $$P=\frac{(2x^2+y+2023)^3}{2024(x^2+y^2)-2024(x+y)+3036}$$ - last post by Danpda47$ Danpda47 11-03-2024
Min $\sum\frac{1}{a^2+1}$ Started by habcy12345, 06-03-2024	3 replies 2502 Views	$Min $\sum\frac{1}{a^2+1}$ - last post by habcy12345$ habcy12345 08-03-2024
$\sum (a+b)^{4}\geq \frac{4}{7}\sum a^{4}$ Started by Hahahahahahahaha, 12-12-2023 bất đẳng thức	4 replies 1353 Views	$$\sum (a+b)^{4}\geq \frac{4}{7}\sum a^{4}$ - last post by Hahahahahahahaha$ Hahahahahahahaha 08-03-2024
$\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{x+2z}{x+z} \geq \frac{5}{3}$ Started by hngmcute, 27-02-2024	1 reply 1176 Views	$$\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{x+2z}{x+z} \geq \frac{5}{3}$ - last post by MHN$ MHN 27-02-2024
$ T = abc.$ Started by 98dfgfdubvh, 26-02-2024	2 replies 1753 Views	dinhvu 27-02-2024
Tìm $GTNN$ của $\left \| 5x^{2}+11xy-5y^{2} \right \|$ Started by Hahahahahahahaha, 23-02-2024	1 reply 2387 Views	$Tìm $GTNN$ của $\left \| 5x^{2}+11xy-5y^{2} \right \|$ - last post by MHN$ MHN 24-02-2024
$$\sqrt[n]{ka+m}+\sqrt[n]{kb+m}+\sqrt[n]{kc+m}$$ Started by nguyenhuybao06, 13-02-2024	4 replies 2856 Views	$$$\sqrt[n]{ka+m}+\sqrt[n]{kb+m}+\sqrt[n]{kc+m}$$ - last post by nguyenhuybao06$ nguyenhuybao06 24-02-2024
Tìm $GTLN$ của: $A=a^2+b^2+c^2-4a-4b-c.$ Started by MHN, 20-02-2024	1 reply 2472 Views	Duc3290 22-02-2024
Answered $\sum\frac{a}{b+c^2}\geq\frac{3}{2}$ Started by nguyenhuybao06, 18-02-2024	7 replies 3896 Views	$$\sum\frac{a}{b+c^2}\geq\frac{3}{2}$ - last post by chuyenndu$ chuyenndu 21-02-2024
Cho $a^2\geq 2b> 0$. Tìm GTNN của $A=\frac{a^6+4b^3}{a^2b^2}$ Started by quanganhvu1503, 19-02-2024	3 replies 3532 Views	$Cho $a^2\geq 2b> 0$. Tìm GTNN của $A=\frac{a^6+4b^3}{a^2b^2}$ - last post by chuyenndu$ chuyenndu 21-02-2024
$GTLN$ $A=\frac{11b^3-a^3}{ab+4b^2}+\frac{11c^3-b^3}{bc+4c^2}+\frac{11a^3-c^3}{ca+4a^2}$ Started by hngmcute, 17-02-2024	2 replies 1480 Views	$$GTLN$ $A=\frac{11b^3-a^3}{ab+4b^2}+\frac{11c^3-b^3}{bc+4c^2}+\frac{11a^3-c^3}{ca+4a^2}$ - last post by Hahahahahahahaha$ Hahahahahahahaha 18-02-2024
$2({a^{m+n}+b^{m+n}})\ge (a^m+b^m)(a^n+b^n)$ Started by hungtannguyen, 17-02-2024	1 reply 1131 Views	$$2({a^{m+n}+b^{m+n}})\ge (a^m+b^m)(a^n+b^n)$ - last post by perfectstrong$ perfectstrong 17-02-2024
$\sum \frac{a_1{}}{({a_1+{a_2+...+a_n{}{}}{}})-{a_1{}}}\geq \frac{n}{n-1}$ Started by Khanh12321, 14-02-2024 bất đẳng thức	5 replies 2152 Views	$$\sum \frac{a_1{}}{({a_1+{a_2+...+a_n{}{}}{}})-{a_1{}}}\geq \frac{n}{n-1}$ - last post by Duc3290$ Duc3290 15-02-2024
Tìm $GTLN$ của tích $3$ số. Started by MHN, 15-02-2024	1 reply 1016 Views	Nguyen Bao Khanh 15-02-2024
$P=\sqrt{a+b^2c^2}+\sqrt{b+c^2a^2}+\sqrt{c+a^2b^2}$ Started by Benben, 18-01-2024	1 reply 1721 Views	$$P=\sqrt{a+b^2c^2}+\sqrt{b+c^2a^2}+\sqrt{c+a^2b^2}$ - last post by nguyenhuybao06$ nguyenhuybao06 15-02-2024
$\frac{x^{3}+y^{3}+z^{3}}{3}\geq xyz+\frac{3}{4}\|(x-y)(y-z)(z-x)\|$ Started by Hahahahahahahaha, 14-02-2024	1 reply 1356 Views	$$\frac{x^{3}+y^{3}+z^{3}}{3}\geq xyz+\frac{3}{4}\|(x-y)(y-z)(z-x)\|$ - last post by nguyenhuybao06$ nguyenhuybao06 14-02-2024
$4\sum \frac{a}{a+b}+\frac{\sum ab^2 + abc}{\sum a^2b + abc} \geq 7$ Started by Duc3290, 13-02-2024	2 replies 1719 Views	$$4\sum \frac{a}{a+b}+\frac{\sum ab^2 + abc}{\sum a^2b + abc} \geq 7$ - last post by nguyenhuybao06$ nguyenhuybao06 14-02-2024
$(a-b)^4+(b-c)^4+(c-a)^4\geq \frac{81}{128}c^4$ Started by MHN, 12-02-2024	0 replies 1080 Views	$$(a-b)^4+(b-c)^4+(c-a)^4\geq \frac{81}{128}c^4$ - last post by MHN$ MHN 12-02-2024
$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\geq \frac{3}{2}.\frac{a^3+b^3+c^3}{a^2+b^2+c^2}$ Started by MHN, 11-02-2024	2 replies 1753 Views	$$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\geq \frac{3}{2}.\frac{a^3+b^3+c^3}{a^2+b^2+c^2}$ - last post by habcy12345$ habcy12345 12-02-2024
Tìm giá trị nhỏ nhât của $A=\frac{2x^2-2xy+9y^2}{x^2+2xy+5y^2}$ Started by phomacsudoi, 09-02-2024	2 replies 1394 Views	$Tìm giá trị nhỏ nhât của $A=\frac{2x^2-2xy+9y^2}{x^2+2xy+5y^2}$ - last post by MHN$ MHN 10-02-2024
$\frac{2c+4a+15}{3b+1}+\frac{4a+3b+15}{2c+1}+\frac{3b+2c+15}{4a+1}\geq 15$ Started by MHN, 08-02-2024	1 reply 1609 Views	$$\frac{2c+4a+15}{3b+1}+\frac{4a+3b+15}{2c+1}+\frac{3b+2c+15}{4a+1}\geq 15$ - last post by habcy12345$ habcy12345 08-02-2024
CMR $\sum \frac{1}{x+yz}\leq \frac{9}{2(xy+yz+zx)}$ Started by DaoTriBach, 31-01-2024	1 reply 1893 Views	$CMR $\sum \frac{1}{x+yz}\leq \frac{9}{2(xy+yz+zx)}$ - last post by Duc3290$ Duc3290 04-02-2024
$\frac{2xy+3}{x^2 +xy + y^2} + \frac{2yz+3}{y^2 +yz + z^2}+ \frac{2zx+3}{z^2 +zx + x^2} \geq 5 $ Started by hovutenha, 07-01-2024	1 reply 1223 Views	$$\frac{2xy+3}{x^2 +xy + y^2} + \frac{2yz+3}{y^2 +yz + z^2}+ \frac{2zx+3}{z^2 +zx + x^2} \geq 5 $ - last post by Duc3290$ Duc3290 29-01-2024
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+3 \ge 2(a+b+c)$ Started by POQ123, 27-01-2024	0 replies 1018 Views	$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+3 \ge 2(a+b+c)$ - last post by POQ123$ POQ123 27-01-2024
min $P=\sum \frac{a^{2}b^{2}}{c(a^{2}+b^{2})}$ Started by Hahahahahahahaha, 25-01-2024 cực trị	1 reply 1993 Views	$min $P=\sum \frac{a^{2}b^{2}}{c(a^{2}+b^{2})}$ - last post by habcy12345$ habcy12345 26-01-2024

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