Toán Lớp 9: Câu 2. a) Chứng minh: (ac + bd)2 + (ad – bc)2 = (a2 + b2)(c2 + d2) b) Chứng minh bất dẳng thức Bunhiacôpxki: (ac + bd)2 ≤ (a2 + b2)(c2

Question

Toán Lớp 9:  Câu 2. a) Chứng minh: (ac + bd)2 + (ad – bc)2 = (a2 + b2)(c2 + d2) b) Chứng minh bất dẳng thức Bunhiacôpxki: (ac + bd)2 ≤ (a2 + b2)(c2 + d2)

chauhoangmy98 · Answer

a)(ac + bd)^2 + (ad &ndash; bc)^2      =a^2c^2 + 2acbd + b^2d^2 + a^2d^2 - 2adbc + b^2c^2     =a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2     =a^2c^2 + a^2d^2+ b^2d^2+ b^2c^2     =a^2(c^2 + d^2) +  b^2(d^2 + c^2)     =(c^2 + d^2)(a^2 + c^2)     &rArr; đpcm       (ad - bc)&sup2; &ge;0   &hArr; (ad)&sup2; - 2abcd + (bc)&sup2; &ge; 0   &hArr; (ad)&sup2; + (bc)&sup2; &ge; 2abcd    &hArr; (ad)&sup2; + (bc)&sup2; + (bd)&sup2; + (ac)&sup2; &ge; 2abcd + (ac)&sup2; + (bd)&sup2;   &hArr;a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2 &ge; 2abcd + (ac)&sup2; + (bd)&sup2;    &hArr; a^2c^2 + a^2d^2+ b^2d^2+ b^2c^2 &ge; 2abcd + (ac)&sup2; + (bd)&sup2;     &hArr; a^2(c^2 + d^2) +  b^2(d^2 + c^2) &ge; 2abcd + (ac)&sup2; + (bd)&sup2;     &hArr; (c^2 + d^2)(a^2 + c^2) &ge; (ac + bd)&sup2;     hay 2abcd + (ac)&sup2; + (bd)&sup2; &le; (c^2 + d^2)(a^2 + c^2)