Các câu hỏi tương tự
Cho \(1\le a\le2;1\le b\le2\)
Chứng minh rằng \(\left(a+b\right)\left(\frac{1}{a}+\frac{1}{b}\right)\le\frac{9}{2}\)
Cho các số thực a, b, c đôi một khác nhau thỏa mãn \(0\le a;b;c\le2\)
CMR: \(\frac{1}{^{\left(a-b\right)^2}}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}\ge\frac{9}{4}\)
cho \(0< a\le\frac{1}{2},0< b\le\frac{1}{2}.CM:\left(\frac{a+b}{2-a-b}\right)^2\ge\frac{ab}{\left(1-a\right)\left(1-b\right)}\)
Cho 2 số thực a, b thay đổi sao cho \(1\le a\le2\) ; \(1\le b\le2\)
Tìm GTLN của biêu thức A= \(\left(a+b^2+\frac{4}{a^2}+\frac{2}{b}\right)\left(b+a^2+\frac{4}{b^2}+\frac{2}{a}\right)\)
cho các số thực a,b,c đôi một khác nhau thỏa mãn \(0\le a,b,c\le2\) . CMR:
\(P=\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}\ge\frac{9}{4}\)
\(P=\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}\)
\(0\le a< b< c\le2\)
GTNN
cho \(1\le a,b,c\le2\)
chúng minh \(\frac{\left(a+b\right)^2}{2c^2+2ab+3c\left(a+b\right)}+\frac{c^2}{\left(a+b\right)^2+6c\left(a+b\right)+4c^2}\ge\frac{3}{11}\)
\(0\le a< b< c\le2.\). Tìm giá trị nhỏ nhất \(P=\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}\)
đặt Pfrac{1}{1-ab}+frac{1}{1-bc}+frac{1}{1-ca}Rightarrow P-3frac{ab}{1-ab}+frac{bc}{1-bc}+frac{ca}{1-ca}lefrac{ab}{1-frac{a^2+b^2}{2}}+frac{bc}{1-frac{b^2+c^2}{2}}+frac{ca}{1-frac{c^2+a^2}{2}}lefrac{1}{2}.frac{left(a+bright)^2}{left(a^2+c^2right)+left(b^2+c^2right)}+frac{1}{2}.frac{left(b+cright)^2}{left(a^2+b^2right)+left(c^2+a^2right)}+frac{1}{2}.frac{left(c+aright)^2}{left(b^2+c^2right)+left(b^2+a^2right)}lefrac{1}{2}.left(frac{a^2}{a^2+c^2}+frac{b^2}{b^2+c^2}+frac{b^2}{a^2+b^2}+frac{c^2}{c^2...
Đọc tiếp
đặt \(P=\frac{1}{1-ab}+\frac{1}{1-bc}+\frac{1}{1-ca}\)
\(\Rightarrow P-3=\frac{ab}{1-ab}+\frac{bc}{1-bc}+\frac{ca}{1-ca}\le\frac{ab}{1-\frac{a^2+b^2}{2}}+\frac{bc}{1-\frac{b^2+c^2}{2}}+\frac{ca}{1-\frac{c^2+a^2}{2}}\)
\(\le\frac{1}{2}.\frac{\left(a+b\right)^2}{\left(a^2+c^2\right)+\left(b^2+c^2\right)}+\frac{1}{2}.\frac{\left(b+c\right)^2}{\left(a^2+b^2\right)+\left(c^2+a^2\right)}+\frac{1}{2}.\frac{\left(c+a\right)^2}{\left(b^2+c^2\right)+\left(b^2+a^2\right)}\)
\(\le\frac{1}{2}.\left(\frac{a^2}{a^2+c^2}+\frac{b^2}{b^2+c^2}+\frac{b^2}{a^2+b^2}+\frac{c^2}{c^2+a^2}+\frac{c^2}{b^2+c^2}+\frac{a^2}{b^2+a^2}\right)=\frac{3}{2}\)
\(\Rightarrow P-3\le\frac{3}{2}\Rightarrow P\le\frac{9}{2}\)