XH
(a^2 + b^2 )( c^2 + d^2 ) - ( ac + bd )^2
= a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 - a^2.c^2 - b^2.d^2 - 2.ab.cd
= a^2.d^2 + b^2.c^2 + 2a.b.c.d
= (ad + bc )^2 >=0
Vậy (a^2 + b^2 )(c^2 + d^2 ) <= ( ........)
Chứng minh với a; b; c; d > 0
\(\sqrt{\left(a^2+c^2\right)\left(b^2+c^2\right)}+\sqrt{\left(a^2+d^2\right)\left(b^2+d^2\right)}\) \(\ge\) \(\left(a+b\right)\left(c+d\right)\)
cho các số thực dương a,b,c,d. Chứng minh rằng: \(\frac{b}{\left(a+\sqrt{b}\right)^2}+\frac{d}{\left(c+\sqrt{d}\right)^2}\ge\frac{\sqrt{bd}}{ac+\sqrt{bd}}\)
chứng minh rằng
\(\left(a+b+c+d\right)^2\ge\frac{8}{3}\left(ab+ac+ad+bc+bd+cd\right)\)
\(\text{Chứng minh: }\)\(\left(ac+bd\right)^2+\left(ad-bc\right)^2=\left(a^2+b^2\right)\left(c^2+d^2\right)\)
Chứng minh:
1) \(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9\)
2) \(\left(ac+bd\right)^{^2}\le\left(a^{^2}+b^{^2}\right)\left(x^{^2}+d^{^2}\right)\)
Bài 1:Cho a,b,c,d là các số dương. Chứng minh rằng :
\(\frac{a^4}{\left(a+b\right)\left(a^2+b^2\right)}+\frac{b^4}{\left(b+c\right)\left(b^2+c^2\right)}+\frac{c^4}{\left(c+d\right)\left(c^2+d^2\right)}+\frac{d^4}{\left(d+a\right)\left(d^2+a^2\right)}\ge\frac{a+b+c+d}{4}\)
Bài 2:Cho \(a>0,b>0,c>0\).\(CM:\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}\ge2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Bài 3: a) Cho x,y,>0. CMR:\(\frac{x^3}{x^2+xy+y^2}\ge\frac{2x-y}{3}\)
b) Chứng minh rằng\(\Sigma\frac{a^3}{a^2+ab+b^2}\ge\frac{a+b+c}{3}\)
Cho a,b,c,d > 0 thỏa mãn \(a^2+b^2+c^2+d^2=1\)
Chứng minh rằng \(\left(1-a\right)\left(1-b\right)\left(1-c\right)\left(1-d\right)\ge abcd\)
Cho ac=bd và ab>0. Chứng minh \(\sqrt{\left(a+b\right)^2+\left(c+d\right)^2}=\sqrt{a^2+d^2}+\sqrt{b^2+c^2}\)
chứng minh \(a^2+b^2+c^2+d^2+e^2\ge a\left(b+c+d+e\right)\)với mọi a,b,c,d,e
