{"id":567229,"date":"2026-07-20T13:49:31","date_gmt":"2026-07-20T06:49:31","guid":{"rendered":"https:\/\/hoanghamobile.com\/tin-tuc\/?p=567229"},"modified":"2026-07-22T15:02:21","modified_gmt":"2026-07-22T08:02:21","slug":"bat-dang-thuc","status":"publish","type":"post","link":"https:\/\/hoanghamobile.com\/tin-tuc\/bat-dang-thuc\/","title":{"rendered":"B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 g\u00ec? T\u00ednh ch\u1ea5t, Cauchy v\u00e0 Bunhiacopxki"},"content":{"rendered":"<p>B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 m\u1ec7nh \u0111\u1ec1 so s\u00e1nh hai bi\u1ec3u th\u1ee9c b\u1eb1ng m\u1ed9t trong c\u00e1c d\u1ea5u: l\u1edbn h\u01a1n (&gt;), nh\u1ecf h\u01a1n (&lt;), l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng (\u2265), nh\u1ecf h\u01a1n ho\u1eb7c b\u1eb1ng (\u2264). V\u00ed d\u1ee5 5 &gt; 3 hay a\u00b2 + b\u00b2 \u2265 2ab \u0111\u1ec1u l\u00e0 b\u1ea5t \u0111\u1eb3ng th\u1ee9c. \u0110\u00e2y l\u00e0 c\u00f4ng c\u1ee5 n\u1ec1n t\u1ea3ng trong To\u00e1n h\u1ecdc, gi\u00fap b\u1ea1n so s\u00e1nh gi\u00e1 tr\u1ecb, t\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t \u2013 nh\u1ecf nh\u1ea5t (c\u1ef1c tr\u1ecb) v\u00e0 ch\u1ee9ng minh nhi\u1ec1u b\u00e0i to\u00e1n kh\u00f3. B\u00e0i vi\u1ebft t\u1ed5ng h\u1ee3p \u0111\u1ecbnh ngh\u0129a, b\u1ea3ng t\u00ednh ch\u1ea5t c\u01a1 b\u1ea3n, hai b\u1ea5t \u0111\u1eb3ng th\u1ee9c kinh \u0111i\u1ec3n Cauchy v\u00e0 Bunhiacopxki, c\u00f9ng lo\u1ea1t v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i chi ti\u1ebft t\u1eebng b\u01b0\u1edbc \u0111\u1ec3 b\u1ea1n n\u1eafm ch\u1eafc c\u00e1ch \u00e1p d\u1ee5ng.<\/p>\n<h2>B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 g\u00ec?<\/h2>\n<p>\u0110\u00e2y l\u00e0 m\u1ed9t m\u1ec7nh \u0111\u1ec1 c\u00f3 d\u1ea1ng a &gt; b, a &lt; b, a \u2265 b ho\u1eb7c a \u2264 b, d\u00f9ng \u0111\u1ec3 so s\u00e1nh hai s\u1ed1 ho\u1eb7c hai bi\u1ec3u th\u1ee9c. M\u1ed9t m\u1ec7nh \u0111\u1ec1 so s\u00e1nh c\u00f3 th\u1ec3 lu\u00f4n \u0111\u00fang (v\u00ed d\u1ee5 x\u00b2 \u2265 0 v\u1edbi m\u1ecdi x) ho\u1eb7c ch\u1ec9 \u0111\u00fang v\u1edbi m\u1ed9t s\u1ed1 gi\u00e1 tr\u1ecb nh\u1ea5t \u0111\u1ecbnh. Trong nhi\u1ec1u b\u00e0i to\u00e1n, ch\u00fang ta c\u1ea7n ch\u1ee9ng minh m\u1ec7nh \u0111\u1ec1 \u0111\u00f3 \u0111\u00fang v\u1edbi m\u1ecdi gi\u00e1 tr\u1ecb c\u1ee7a bi\u1ebfn ho\u1eb7c t\u00ecm \u0111i\u1ec1u ki\u1ec7n \u0111\u1ec3 n\u00f3 x\u1ea3y ra.<\/p>\n<p>Trong b\u1ea5t \u0111\u1eb3ng th\u1ee9c a &gt; b, a \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 v\u1ebf tr\u00e1i, b l\u00e0 v\u1ebf ph\u1ea3i. D\u00f9ng d\u1ea5u &gt; ho\u1eb7c &lt; l\u00e0 b\u1ea5t \u0111\u1eb3ng th\u1ee9c nghi\u00eam ng\u1eb7t (kh\u00f4ng c\u00f3 d\u1ea5u b\u1eb1ng); d\u00f9ng d\u1ea5u \u2265 ho\u1eb7c \u2264 l\u00e0 b\u1ea5t \u0111\u1eb3ng th\u1ee9c kh\u00f4ng nghi\u00eam ng\u1eb7t (c\u00f3 th\u1ec3 c\u00f3 d\u1ea5u b\u1eb1ng) \u2014 ph\u00e2n bi\u1ec7t hai lo\u1ea1i n\u00e0y gi\u00fap b\u1ea1n x\u00e1c \u0111\u1ecbnh \u0111\u00fang \u0111i\u1ec1u ki\u1ec7n c\u1ef1c tr\u1ecb.<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"size-full wp-image-567726 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-1.jpg\" alt=\"bat-dang-thuc-1\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-1.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-1-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-1-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>C\u00e1c t\u00ednh ch\u1ea5t c\u01a1 b\u1ea3n c\u1ee7a b\u1ea5t \u0111\u1eb3ng th\u1ee9c<\/h2>\n<p>B\u1ea5t \u0111\u1eb3ng th\u1ee9c c\u00f3 nhi\u1ec1u t\u00ednh ch\u1ea5t quan tr\u1ecdng, d\u00f9ng \u0111\u1ec3 bi\u1ebfn \u0111\u1ed5i v\u00e0 ch\u1ee9ng minh; b\u1ea3ng d\u01b0\u1edbi \u0111\u00e2y t\u1ed5ng h\u1ee3p c\u00e1c t\u00ednh ch\u1ea5t c\u01a1 b\u1ea3n nh\u1ea5t.<\/p>\n<table>\n<thead>\n<tr>\n<th>T\u00ednh ch\u1ea5t<\/th>\n<th>Ph\u00e1t bi\u1ec3u<\/th>\n<th>V\u00ed d\u1ee5 minh h\u1ecda<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>B\u1eafc c\u1ea7u<\/td>\n<td>N\u1ebfu a &gt; b v\u00e0 b &gt; c th\u00ec a &gt; c<\/td>\n<td>5 &gt; 3 v\u00e0 3 &gt; 1 n\u00ean 5 &gt; 1<\/td>\n<\/tr>\n<tr>\n<td>C\u1ed9ng hai v\u1ebf v\u1edbi c\u00f9ng m\u1ed9t s\u1ed1<\/td>\n<td>N\u1ebfu a &gt; b th\u00ec a + c &gt; b + c (chi\u1ec1u kh\u00f4ng \u0111\u1ed5i)<\/td>\n<td>2 &gt; 1 n\u00ean 2 + 5 &gt; 1 + 5, t\u1ee9c 7 &gt; 6<\/td>\n<\/tr>\n<tr>\n<td>Nh\u00e2n hai v\u1ebf v\u1edbi s\u1ed1 d\u01b0\u01a1ng<\/td>\n<td>N\u1ebfu a &gt; b v\u00e0 c &gt; 0 th\u00ec ac &gt; bc (gi\u1eef chi\u1ec1u)<\/td>\n<td>3 &gt; 2, nh\u00e2n 4: 12 &gt; 8<\/td>\n<\/tr>\n<tr>\n<td>Nh\u00e2n hai v\u1ebf v\u1edbi s\u1ed1 \u00e2m<\/td>\n<td>N\u1ebfu a &gt; b v\u00e0 c &lt; 0 th\u00ec ac &lt; bc (\u0111\u1ed5i chi\u1ec1u)<\/td>\n<td>3 &gt; 2, nh\u00e2n \u22121: \u22123 &lt; \u22122<\/td>\n<\/tr>\n<tr>\n<td>C\u1ed9ng v\u1ebf theo v\u1ebf<\/td>\n<td>N\u1ebfu a &gt; b v\u00e0 c &gt; d th\u00ec a + c &gt; b + d<\/td>\n<td>5 &gt; 3 v\u00e0 4 &gt; 1 n\u00ean 9 &gt; 4<\/td>\n<\/tr>\n<tr>\n<td>Ngh\u1ecbch \u0111\u1ea3o (s\u1ed1 d\u01b0\u01a1ng)<\/td>\n<td>N\u1ebfu a &gt; b &gt; 0 th\u00ec 1\/a &lt; 1\/b<\/td>\n<td>4 &gt; 2 &gt; 0 n\u00ean 1\/4 &lt; 1\/2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>L\u01b0u \u00fd: nh\u00e2n v\u1edbi s\u1ed1 \u00e2m l\u00e0 l\u1ed7i d\u1ec5 nh\u1ea7m nh\u1ea5t \u2014 lu\u00f4n nh\u1edb \u0111\u1ed5i chi\u1ec1u d\u1ea5u khi nh\u00e2n ho\u1eb7c chia c\u1ea3 hai v\u1ebf cho m\u1ed9t s\u1ed1 \u00e2m.<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-567727 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-2.jpg\" alt=\"bat-dang-thuc-2\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-2.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-2-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-2-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>B\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy (AM \u2013 GM)<\/h2>\n<p>\u0110\u00e2y l\u00e0 b\u1ea5t \u0111\u1eb3ng th\u1ee9c n\u1ed5i ti\u1ebfng v\u00e0 \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng nhi\u1ec1u nh\u1ea5t trong ch\u01b0\u01a1ng tr\u00ecnh To\u00e1n ph\u1ed5 th\u00f4ng. V\u1edbi hai s\u1ed1 kh\u00f4ng \u00e2m a, b:<\/p>\n<p>(a + b)\/2 \u2265 \u221a(ab) (trung b\u00ecnh c\u1ed9ng lu\u00f4n l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng trung b\u00ecnh nh\u00e2n).<\/p>\n<p>D\u1ea5u b\u1eb1ng x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi a = b. C\u00f3 th\u1ec3 ch\u1ee9ng minh nhanh b\u1ea5t \u0111\u1eb3ng th\u1ee9c n\u00e0y b\u1eb1ng bi\u1ebfn \u0111\u1ed5i t\u01b0\u01a1ng \u0111\u01b0\u01a1ng: v\u00ec (\u221aa \u2212 \u221ab)\u00b2 \u2265 0 v\u1edbi m\u1ecdi a, b \u2265 0, khai tri\u1ec3n \u0111\u01b0\u1ee3c a \u2212 2\u221a(ab) + b \u2265 0, suy ra a + b \u2265 2\u221a(ab), t\u1ee9c (a + b)\/2 \u2265 \u221a(ab). D\u1ea5u b\u1eb1ng x\u1ea3y ra khi \u221aa = \u221ab, hay a = b.<\/p>\n<p>M\u1edf r\u1ed9ng cho ba s\u1ed1 kh\u00f4ng \u00e2m a, b, c: (a + b + c)\/3 \u2265 \u221b(abc), d\u1ea5u b\u1eb1ng khi a = b = c. B\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy th\u01b0\u1eddng d\u00f9ng \u0111\u1ec3 t\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t, nh\u1ecf nh\u1ea5t c\u1ee7a m\u1ed9t bi\u1ec3u th\u1ee9c khi t\u1ed5ng ho\u1eb7c t\u00edch c\u1ee7a c\u00e1c s\u1ed1 kh\u00f4ng \u0111\u1ed5i.<\/p>\n<p><strong>Xem th\u00eam: <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/dien-tich-hinh-tron\/\">C\u00f4ng Th\u1ee9c T\u00ednh Di\u1ec7n T\u00edch H\u00ecnh Tr\u00f2n: C\u00e1ch T\u00ednh, V\u00ed D\u1ee5 &amp; B\u00e0i T\u1eadp<\/a><\/strong><\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-567728 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-3.jpg\" alt=\"bat-dang-thuc-3\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-3.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-3-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-3-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>B\u1ea5t \u0111\u1eb3ng th\u1ee9c Bunhiacopxki<\/h2>\n<p>V\u1edbi c\u00e1c s\u1ed1 th\u1ef1c a, b, x, y, b\u1ea5t \u0111\u1eb3ng th\u1ee9c Bunhiacopxki (c\u00f2n g\u1ecdi l\u00e0 Cauchy-Schwarz) ph\u00e1t bi\u1ec3u:<\/p>\n<p>(a\u00b2 + b\u00b2)(x\u00b2 + y\u00b2) \u2265 (ax + by)\u00b2.<\/p>\n<p>D\u1ea5u b\u1eb1ng x\u1ea3y ra khi a\/x = b\/y (v\u1edbi x, y kh\u00e1c 0). \u0110\u00e2y l\u00e0 c\u00f4ng c\u1ee5 m\u1ea1nh \u0111\u1ec3 ch\u1ee9ng minh nhi\u1ec1u b\u1ea5t \u0111\u1eb3ng th\u1ee9c ph\u1ee9c t\u1ea1p v\u00e0 t\u00ecm c\u1ef1c tr\u1ecb c\u1ee7a bi\u1ec3u th\u1ee9c.<\/p>\n<p>C\u00f3 th\u1ec3 ki\u1ec3m ch\u1ee9ng b\u1ea5t \u0111\u1eb3ng th\u1ee9c n\u00e0y b\u1eb1ng c\u00e1ch x\u00e9t hi\u1ec7u hai v\u1ebf: (a\u00b2 + b\u00b2)(x\u00b2 + y\u00b2) \u2212 (ax + by)\u00b2 = a\u00b2y\u00b2 \u2212 2abxy + b\u00b2x\u00b2 = (ay \u2212 bx)\u00b2. V\u00ec b\u00ecnh ph\u01b0\u01a1ng c\u1ee7a m\u1ed9t s\u1ed1 th\u1ef1c lu\u00f4n kh\u00f4ng \u00e2m n\u00ean (ay \u2212 bx)\u00b2 \u2265 0, do \u0111\u00f3 b\u1ea5t \u0111\u1eb3ng th\u1ee9c tr\u00ean lu\u00f4n \u0111\u00fang. D\u1ea5u b\u1eb1ng x\u1ea3y ra khi ay = bx, t\u1ee9c a\/x = b\/y.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-567729 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-4.jpg\" alt=\"bat-dang-thuc-4\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-4.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-4-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-4-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>V\u00ed d\u1ee5 v\u1ec1 b\u1ea5t \u0111\u1eb3ng th\u1ee9c (c\u00f3 l\u1eddi gi\u1ea3i t\u1eebng b\u01b0\u1edbc)<\/h2>\n<p>Ph\u1ea7n d\u01b0\u1edbi \u0111\u00e2y tr\u00ecnh b\u00e0y m\u1ed9t s\u1ed1 v\u00ed d\u1ee5 ti\u00eau bi\u1ec3u, c\u00f3 l\u1eddi gi\u1ea3i chi ti\u1ebft t\u1eebng b\u01b0\u1edbc, gi\u00fap b\u1ea1n h\u00ecnh dung c\u00e1ch \u00e1p d\u1ee5ng c\u00e1c t\u00ednh ch\u1ea5t v\u00e0 hai b\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy, Bunhiacopxki \u1edf tr\u00ean.<\/p>\n<ul>\n<li><strong>V\u00ed d\u1ee5 1:<\/strong> Ch\u1ee9ng minh a\u00b2 + b\u00b2 \u2265 2ab v\u1edbi m\u1ecdi a, b. L\u1eddi gi\u1ea3i: a\u00b2 + b\u00b2 \u2212 2ab = (a \u2212 b)\u00b2 \u2265 0 (b\u00ecnh ph\u01b0\u01a1ng lu\u00f4n kh\u00f4ng \u00e2m), suy ra a\u00b2 + b\u00b2 \u2265 2ab; d\u1ea5u b\u1eb1ng khi a = b.<\/li>\n<li><strong>V\u00ed d\u1ee5 2:<\/strong> Cho a, b &gt; 0, ab = 16. T\u00ecm gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a a + b. L\u1eddi gi\u1ea3i: Theo Cauchy, a + b \u2265 2\u221a(ab) = 2\u221a16 = 8; d\u1ea5u b\u1eb1ng khi a = b, k\u1ebft h\u1ee3p ab = 16 suy ra a = b = 4. V\u1eady gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a a + b l\u00e0 8, t\u1ea1i a = b = 4.<\/li>\n<li><strong>V\u00ed d\u1ee5 3:<\/strong> So s\u00e1nh 3\u00b2 + 4\u00b2 v\u00e0 2 \u00b7 3 \u00b7 4. L\u1eddi gi\u1ea3i: 3\u00b2 + 4\u00b2 = 25; 2 \u00b7 3 \u00b7 4 = 24. V\u00ec 25 &gt; 24 n\u00ean 3\u00b2 + 4\u00b2 &gt; 2 \u00b7 3 \u00b7 4, \u0111\u00fang nh\u01b0 a\u00b2 + b\u00b2 \u2265 2ab (3 \u2260 4 n\u00ean kh\u00f4ng c\u00f3 d\u1ea5u b\u1eb1ng).<\/li>\n<li><strong>V\u00ed d\u1ee5 4:<\/strong> Cho x, y &gt; 0, x + y = 10. T\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a xy. L\u1eddi gi\u1ea3i: Theo Cauchy, (x + y)\/2 \u2265 \u221a(xy) n\u00ean 5 \u2265 \u221a(xy), b\u00ecnh ph\u01b0\u01a1ng \u0111\u01b0\u1ee3c 25 \u2265 xy; d\u1ea5u b\u1eb1ng khi x = y = 5. V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a xy l\u00e0 25.<\/li>\n<li><strong>V\u00ed d\u1ee5 5:<\/strong> Cho a\u00b2 + b\u00b2 = 4. T\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a P = a + b. L\u1eddi gi\u1ea3i: Theo Bunhiacopxki v\u1edbi x = y = 1: (a\u00b2 + b\u00b2)(1\u00b2 + 1\u00b2) \u2265 (a + b)\u00b2, t\u1ee9c 8 \u2265 (a + b)\u00b2, n\u00ean (a + b)\u00b2 \u2264 8; d\u1ea5u b\u1eb1ng khi a = b, k\u1ebft h\u1ee3p a\u00b2 + b\u00b2 = 4 suy ra a = b = \u221a2. V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a P l\u00e0 2\u221a2.<\/li>\n<li><strong>V\u00ed d\u1ee5 6:<\/strong> Cho x &gt; 5 v\u00e0 5 &gt; y. So s\u00e1nh x v\u00e0 y. L\u1eddi gi\u1ea3i: Theo t\u00ednh ch\u1ea5t b\u1eafc c\u1ea7u, t\u1eeb x &gt; 5 v\u00e0 5 &gt; y suy ra x &gt; y.<\/li>\n<\/ul>\n<p><strong>Xem th\u00eam: <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-tinh-quang-duong\/\">C\u00f4ng Th\u1ee9c T\u00ednh Qu\u00e3ng \u0110\u01b0\u1eddng: C\u00e1ch T\u00ednh To\u00e1n L\u1edbp 5 &amp; V\u1eadt L\u00fd L\u1edbp 8<\/a><\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-567730 aligncenter\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-5.jpg\" alt=\"bat-dang-thuc-5\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-5.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-5-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/07\/bat-dang-thuc-5-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>C\u00e2u h\u1ecfi th\u01b0\u1eddng g\u1eb7p v\u1ec1 b\u1ea5t \u0111\u1eb3ng th\u1ee9c<\/h2>\n<h3>B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 g\u00ec?<\/h3>\n<p>L\u00e0 m\u1ec7nh \u0111\u1ec1 so s\u00e1nh hai bi\u1ec3u th\u1ee9c b\u1eb1ng m\u1ed9t trong c\u00e1c d\u1ea5u &gt;, &lt;, \u2265 ho\u1eb7c \u2264.<\/p>\n<h3>B\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy (AM \u2013 GM) ph\u00e1t bi\u1ec3u th\u1ebf n\u00e0o?<\/h3>\n<p>V\u1edbi hai s\u1ed1 kh\u00f4ng \u00e2m a, b: (a + b)\/2 \u2265 \u221a(ab), d\u1ea5u b\u1eb1ng x\u1ea3y ra khi a = b.<\/p>\n<h3>B\u1ea5t \u0111\u1eb3ng th\u1ee9c Bunhiacopxki ph\u00e1t bi\u1ec3u th\u1ebf n\u00e0o?<\/h3>\n<p>V\u1edbi c\u00e1c s\u1ed1 th\u1ef1c a, b, x, y: (a\u00b2 + b\u00b2)(x\u00b2 + y\u00b2) \u2265 (ax + by)\u00b2, d\u1ea5u b\u1eb1ng x\u1ea3y ra khi a\/x = b\/y.<\/p>\n<h3>Khi n\u00e0o b\u1ea5t \u0111\u1eb3ng th\u1ee9c \u0111\u1ed5i chi\u1ec1u?<\/h3>\n<p>Khi nh\u00e2n ho\u1eb7c chia c\u1ea3 hai v\u1ebf c\u1ee7a b\u1ea5t \u0111\u1eb3ng th\u1ee9c v\u1edbi c\u00f9ng m\u1ed9t s\u1ed1 \u00e2m.<\/p>\n<h3>B\u1ea5t \u0111\u1eb3ng th\u1ee9c a\u00b2 + b\u00b2 \u2265 2ab c\u00f3 \u0111\u00fang v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c a, b kh\u00f4ng?<\/h3>\n<p>\u0110\u00fang, v\u00ec a\u00b2 + b\u00b2 \u2212 2ab = (a \u2212 b)\u00b2 \u2265 0 v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c a, b.<\/p>\n<h3>B\u1ea5t \u0111\u1eb3ng th\u1ee9c kh\u00e1c b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u1edf \u0111i\u1ec3m n\u00e0o?<\/h3>\n<p>B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 m\u1ec7nh \u0111\u1ec1 so s\u00e1nh n\u00f3i chung; b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 b\u1ea5t \u0111\u1eb3ng th\u1ee9c ch\u1ee9a \u1ea9n, c\u1ea7n t\u00ecm t\u1eadp gi\u00e1 tr\u1ecb c\u1ee7a \u1ea9n \u0111\u1ec3 m\u1ec7nh \u0111\u1ec1 \u0111\u00f3 \u0111\u00fang.<\/p>\n<h3>D\u1ea5u b\u1eb1ng trong b\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy v\u00e0 Bunhiacopxki c\u00f3 \u00fd ngh\u0129a g\u00ec?<\/h3>\n<p>D\u1ea5u b\u1eb1ng cho bi\u1ebft \u0111i\u1ec1u ki\u1ec7n \u0111\u1ec3 bi\u1ec3u th\u1ee9c \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t ho\u1eb7c nh\u1ecf nh\u1ea5t (c\u1ef1c tr\u1ecb).<\/p>\n<h3>H\u1ecdc b\u1ea5t \u0111\u1eb3ng th\u1ee9c \u0111\u1ec3 l\u00e0m g\u00ec?<\/h3>\n<p>B\u1ea5t \u0111\u1eb3ng th\u1ee9c gi\u00fap so s\u00e1nh, ch\u1ee9ng minh v\u00e0 t\u00ecm c\u1ef1c tr\u1ecb c\u1ee7a bi\u1ec3u th\u1ee9c; \u0111\u00e2y c\u0169ng l\u00e0 n\u1ec1n t\u1ea3ng \u0111\u1ec3 h\u1ecdc b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh v\u00e0 kh\u1ea3o s\u00e1t h\u00e0m s\u1ed1 \u1edf l\u1edbp tr\u00ean.<\/p>\n<h2>K\u1ebft lu\u1eadn<\/h2>\n<p>T\u00f3m l\u1ea1i, b\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 m\u1ec7nh \u0111\u1ec1 so s\u00e1nh hai bi\u1ec3u th\u1ee9c b\u1eb1ng c\u00e1c d\u1ea5u &gt;, &lt;, \u2265 ho\u1eb7c \u2264, c\u00f9ng nh\u1eefng t\u00ednh ch\u1ea5t c\u01a1 b\u1ea3n nh\u01b0 b\u1eafc c\u1ea7u, c\u1ed9ng v\u00e0 nh\u00e2n hai v\u1ebf. Hai b\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy v\u00e0 Bunhiacopxki l\u00e0 c\u00f4ng c\u1ee5 m\u1ea1nh \u0111\u1ec3 ch\u1ee9ng minh v\u00e0 t\u00ecm c\u1ef1c tr\u1ecb, c\u00f2n \u0111i\u1ec1u ki\u1ec7n d\u1ea5u b\u1eb1ng lu\u00f4n l\u00e0 ch\u00eca kh\u00f3a c\u1ee7a b\u00e0i to\u00e1n. M\u1eddi b\u1ea1n tham kh\u1ea3o th\u00eam c\u00e1c b\u00e0i<strong> <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-to-hop-chinh-hop\/\">c\u00f4ng th\u1ee9c ho\u00e1n v\u1ecb, ch\u1ec9nh h\u1ee3p, t\u1ed5 h\u1ee3p<\/a>, <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/nhi-thuc-newton\/\">nh\u1ecb th\u1ee9c Newton<\/a>, <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/tap-hop-so-nguyen-z-la-gi-giai-ma-toan-dien-ve-z-trong-toan-hoc\/\">t\u1eadp h\u1ee3p s\u1ed1 nguy\u00ean Z<\/a> <\/strong>v\u00e0 <strong><a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/dien-tich-tam-giac\/\">di\u1ec7n t\u00edch tam gi\u00e1c<\/a> <\/strong>tr\u00ean chuy\u00ean m\u1ee5c tin t\u1ee9c c\u1ee7a Ho\u00e0ng H\u00e0 Mobile.<br \/>\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 g\u00ec?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"L\u00e0 m\u1ec7nh \u0111\u1ec1 so s\u00e1nh hai bi\u1ec3u th\u1ee9c b\u1eb1ng m\u1ed9t trong c\u00e1c d\u1ea5u >, <, \u2265 ho\u1eb7c \u2264.\"}},{\"@type\":\"Question\",\"name\":\"B\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy (AM \u2013 GM) ph\u00e1t bi\u1ec3u th\u1ebf n\u00e0o?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"V\u1edbi hai s\u1ed1 kh\u00f4ng \u00e2m a, b: (a + b)\/2 \u2265 \u221a(ab), d\u1ea5u b\u1eb1ng x\u1ea3y ra khi a = b.\"}},{\"@type\":\"Question\",\"name\":\"B\u1ea5t \u0111\u1eb3ng th\u1ee9c Bunhiacopxki ph\u00e1t bi\u1ec3u th\u1ebf n\u00e0o?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"V\u1edbi c\u00e1c s\u1ed1 th\u1ef1c a, b, x, y: (a\u00b2 + b\u00b2)(x\u00b2 + y\u00b2) \u2265 (ax + by)\u00b2, d\u1ea5u b\u1eb1ng x\u1ea3y ra khi a\/x = b\/y.\"}},{\"@type\":\"Question\",\"name\":\"Khi n\u00e0o b\u1ea5t \u0111\u1eb3ng th\u1ee9c \u0111\u1ed5i chi\u1ec1u?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Khi nh\u00e2n ho\u1eb7c chia c\u1ea3 hai v\u1ebf c\u1ee7a b\u1ea5t \u0111\u1eb3ng th\u1ee9c v\u1edbi c\u00f9ng m\u1ed9t s\u1ed1 \u00e2m.\"}},{\"@type\":\"Question\",\"name\":\"B\u1ea5t \u0111\u1eb3ng th\u1ee9c a\u00b2 + b\u00b2 \u2265 2ab c\u00f3 \u0111\u00fang v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c a, b kh\u00f4ng?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"\u0110\u00fang, v\u00ec a\u00b2 + b\u00b2 \u2212 2ab = (a \u2212 b)\u00b2 \u2265 0 v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c a, b.\"}},{\"@type\":\"Question\",\"name\":\"B\u1ea5t \u0111\u1eb3ng th\u1ee9c kh\u00e1c b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u1edf \u0111i\u1ec3m n\u00e0o?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 m\u1ec7nh \u0111\u1ec1 so s\u00e1nh n\u00f3i chung; b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 b\u1ea5t \u0111\u1eb3ng th\u1ee9c ch\u1ee9a \u1ea9n, c\u1ea7n t\u00ecm t\u1eadp gi\u00e1 tr\u1ecb c\u1ee7a \u1ea9n \u0111\u1ec3 m\u1ec7nh \u0111\u1ec1 \u0111\u00f3 \u0111\u00fang.\"}},{\"@type\":\"Question\",\"name\":\"D\u1ea5u b\u1eb1ng trong b\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy v\u00e0 Bunhiacopxki c\u00f3 \u00fd ngh\u0129a g\u00ec?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"D\u1ea5u b\u1eb1ng cho bi\u1ebft \u0111i\u1ec1u ki\u1ec7n \u0111\u1ec3 bi\u1ec3u th\u1ee9c \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t ho\u1eb7c nh\u1ecf nh\u1ea5t (c\u1ef1c tr\u1ecb).\"}},{\"@type\":\"Question\",\"name\":\"H\u1ecdc b\u1ea5t \u0111\u1eb3ng th\u1ee9c \u0111\u1ec3 l\u00e0m g\u00ec?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"B\u1ea5t \u0111\u1eb3ng th\u1ee9c gi\u00fap so s\u00e1nh, ch\u1ee9ng minh v\u00e0 t\u00ecm c\u1ef1c tr\u1ecb c\u1ee7a bi\u1ec3u th\u1ee9c; \u0111\u00e2y c\u0169ng l\u00e0 n\u1ec1n t\u1ea3ng \u0111\u1ec3 h\u1ecdc b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh v\u00e0 kh\u1ea3o s\u00e1t h\u00e0m s\u1ed1 \u1edf l\u1edbp tr\u00ean.\"}}]}\n<\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 m\u1ec7nh \u0111\u1ec1 so s\u00e1nh hai bi\u1ec3u th\u1ee9c b\u1eb1ng m\u1ed9t trong c\u00e1c d\u1ea5u: l\u1edbn h\u01a1n (&gt;), nh\u1ecf h\u01a1n (&lt;), l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng (\u2265), nh\u1ecf h\u01a1n ho\u1eb7c b\u1eb1ng (\u2264). V\u00ed d\u1ee5 5 &gt; 3 hay a\u00b2 + b\u00b2 \u2265 2ab \u0111\u1ec1u l\u00e0 b\u1ea5t \u0111\u1eb3ng th\u1ee9c. \u0110\u00e2y l\u00e0 c\u00f4ng c\u1ee5 n\u1ec1n t\u1ea3ng trong [&hellip;]<\/p>\n","protected":false},"author":65,"featured_media":567725,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"b\u1ea5t \u0111\u1eb3ng th\u1ee9c","_yoast_wpseo_title":"B\u1ea5t \u0111\u1eb3ng th\u1ee9c: t\u00ednh ch\u1ea5t v\u00e0 v\u00ed d\u1ee5","_yoast_wpseo_metadesc":"B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 g\u00ec? \u0110\u1ecbnh ngh\u0129a, b\u1ea3ng t\u00ednh ch\u1ea5t, B\u0110T Cauchy v\u00e0 Bunhiacopxki k\u00e8m v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i chi ti\u1ebft, d\u1ec5 hi\u1ec3u. Xem ngay t\u1ea1i Ho\u00e0ng H\u00e0 Mobile.","footnotes":""},"categories":[16],"tags":[],"class_list":["post-567229","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-thu-thuat"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.3 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>B\u1ea5t \u0111\u1eb3ng th\u1ee9c: t\u00ednh ch\u1ea5t v\u00e0 v\u00ed d\u1ee5<\/title>\n<meta name=\"description\" content=\"B\u1ea5t \u0111\u1eb3ng th\u1ee9c l\u00e0 g\u00ec? \u0110\u1ecbnh ngh\u0129a, b\u1ea3ng t\u00ednh ch\u1ea5t, B\u0110T Cauchy v\u00e0 Bunhiacopxki k\u00e8m v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i chi ti\u1ebft, d\u1ec5 hi\u1ec3u. 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