\(\Leftrightarrow a\left(-\dfrac{8}{a^2+16}+\dfrac{a+8}{2a^2+16}\right)>=0\)
=>\(\dfrac{a^2\left(a-4\right)^2}{\left(a^2+16\right)\left(2a^2+16\right)}>=0\)(luôn đúng)
CMR: \(\dfrac{1}{\left(1+a\right)^2}+\dfrac{1}{\left(1+b\right)^2}\ge\dfrac{1}{1+ab}\forall a,b\ge0\)
usechatgpt init success
Cho \(x,y>0;x+y=1\) . Tìm Min \(P=\left(x^2+\dfrac{1}{y^2}\right)\left(y^2+\dfrac{1}{x^2}\right)-\dfrac{17}{6}\)
usechatgpt init success
\(1,Cho.a,b,c\ge1.CMR:\left(a-\dfrac{1}{b}\right)\left(b-\dfrac{1}{c}\right)\left(c-\dfrac{1}{a}\right)\ge\left(a-\dfrac{1}{a}\right)\left(b-\dfrac{1}{b}\right)\left(c-\dfrac{1}{c}\right)\)
2, Cho a,b,c>0.CMR:
\(\dfrac{a+b}{bc+a^2}+\dfrac{b+c}{ac+b^2}+\dfrac{c+a}{ab+c^2}\le\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
CMR:
a,\(\dfrac{a^2+b^2}{2}\ge\dfrac{\left(a+b\right)^2}{2}\)
b,Cho a+b=1,a>0,b>0 CMR:\(\left(1+\dfrac{1}{a}\right)\left(1+\dfrac{1}{b}\right)\)\(\ge9\)
B1: Cho \(0\le a,b,c\le2\) thỏa mãn \(a+b+c=3\). CMR: \(a^2+b^2+c^2\le5\)
B2: Cho \(a,b\ge0\) thỏa mãn \(a^2+b^2=a+b\). TÌm GTLN \(S=\dfrac{a}{a+1}+\dfrac{b}{b+1}\)
B3: CMR: \(\dfrac{1}{\left(x-y\right)^2}+\dfrac{1}{x^2}+\dfrac{1}{y^2}\ge\dfrac{4}{xy}\forall x\ne y,xy\ne0\)
Sử dụng BĐT Bunhiacopxki cộng mẫu, lm bài toán sau:
Cho a,b,c là các số thực dương. CMR:
\(\dfrac{2\left(b+c-a\right)^2}{2a^2+\left(b+c\right)^2}+\dfrac{2\left(c+a-b\right)^2}{2b^2+\left(c+a\right)^2}+\dfrac{2\left(a+b-c\right)^2}{2c^2+\left(a+b\right)^2}\ge1\)
Chứng minh rằng: \(\left(a^2+b^2+c^2\right)\left[\left(\dfrac{1}{\left(a-b\right)^2}+\dfrac{1}{\left(b-c\right)^2}+\dfrac{1}{\left(c-a\right)^2}\right)\right]\ge\dfrac{9}{2}\)
cho
\(A=\dfrac{1}{2}+\dfrac{3}{2}+\left(\dfrac{3}{2}\right)^2+\left(\dfrac{3}{2}\right)^3+\left(\dfrac{3}{2}\right)^4+...+\left(\dfrac{3}{2}\right)^{2021}\)
\(B=\left(\dfrac{3}{2}\right)^{2013}:2\)
tính B-A
