\(\sqrt{1-\left(\left|x\right|-1\right)^2}\)
Tìm a,b nguyên biết: \(2a^2+2b^2+2ab-8a-8b+10=0\)
\(P=\frac{a}{\sqrt{\left(b+1\right)\left(b^2-b+1\right)}}+\frac{b}{\sqrt{\left(c+1\right)\left(c^2-c+1\right)}}+\frac{c}{\sqrt{\left(a+1\right)\left(a^2-a+1\right)}}\)
\(\ge\frac{2a}{b^2+2}+\frac{2b}{c^2+2}+\frac{2c}{a^2+2}=\left(a+b+c\right)-\left(\frac{ab^2}{b^2+2}+\frac{bc^2}{c^2+2}+\frac{ca^2}{a^2+2}\right)\)
\(=6-\left(\frac{2ab^2}{b^2+4+b^2}+\frac{2bc^2}{c^2+4+c^2}+\frac{2ca^2}{a^2+4+a^2}\right)\ge6-\left(\frac{2ab}{b+4}+\frac{2bc}{c+4}+\frac{2ca}{a+4}\right)\)
\(=6-\left(2a+2b+2c-\frac{8a}{b+4}-\frac{8b}{c+4}-\frac{8c}{a+4}\right)\)
\(=\frac{8a}{b+4}+\frac{8b}{c+4}+\frac{8c}{a+4}-6=\frac{8a^2}{ab+4a}+\frac{8b^2}{bc+4b}+\frac{8c^2}{ca+4c}-6\)
\(\ge\frac{8\left(a+b+c\right)^2}{\left(ab+bc+ca\right)+4\left(a+b+c\right)}-6\ge\frac{288}{\frac{\left(a+b+c\right)^2}{3}+24}-6=2\)
tìm a, b để hệ phương trình sau có nghiệm
\(\hept{\begin{cases}\left(2a+b+1\right)x+\left(a-2b-2\right)y=5a\\\left(3a^2+4b^2+2\right)x+\left(2a^2-8b^2-4\right)y=8a^2\end{cases}}\)
cho a, b >0. hãy đơn giản biểu thức \(\frac{\sqrt{a^{3^{ }}+2a^2b}+\sqrt{a^4+2ab}-\sqrt{a^3}-a^2b}{\sqrt{\left(2a+b-\sqrt{a^2+2ab}\right)}.\left(\sqrt[3]{a^2}-\sqrt[6]{a^5}+a\right)}\)
Vì a>0; b>0 nên a + b \geq 4ab1+ab4ab1+ab
\Leftrightarrow (a + b)(1 + ab)\geq 4ab
\Leftrightarrow a + b + a^2b+ab^2\geq 4ab
\Leftrightarrow a + b + a^b + ab^2 - 4ab\geq 0
\Leftrightarrow (a^2b - 2ab + b) + (ab^2 - 2ab +a) \geq 0
\Leftrightarrow b(a^2 -2a + 1) + a(b^2 - 2B + 1)\geq 0
\Leftrightarrow b(a-1)^2 + a(b-1)^2\geq 0
\Rightarrow Bất đẳng thức đúng\Rightarrow đpcm.
cho a,b,c >0 hãy đơn giản bt :
A=\(\frac{\sqrt{a^3+2a^2b}+\sqrt{a^4+2a^3b}-\sqrt{a^3}-a^2b}{\sqrt{2a+b-\sqrt{a^2+2ab}}.\left(\sqrt[3]{a^2}-\sqrt[6]{a^5}+a\right)}\)
Cho a,b > 0. Hãy đơn giản biểu thức :
\(T=\frac{\sqrt{a^3+2a^2b}+\sqrt{a^4+2a^3b}-\sqrt{a^3}-a^2b}{\sqrt{\left(2a+b-\sqrt{a^2+2ab}\right)}.\left(\sqrt[3]{a^2}-\sqrt[6]{a^5}+a\right)}\)
Bài toán : Cho a,b > 0
CMR : \(\frac{a^2b}{2a^3+b^3}+\frac{2}{3}\ge\frac{a^2+2ab}{2a^2+b^2}\)
Cho a,b >0 va a +b +2ab = 12. Tim min cua P =\(\frac{a^2+ab}{2b+a}\)+\(\frac{b^2+ab}{2a+b}\)
