Áp dụng BĐT AM-GM ta có:
\(\left(\frac{12}{5}\right)^x+\left(\frac{15}{4}\right)^x\ge2\sqrt{9^x}=2\cdot3^x\)
\(\left(\frac{15}{4}\right)^x+\left(\frac{20}{3}\right)^x\ge2\sqrt{25^x}=2\cdot5^x\)
\(\left(\frac{20}{3}\right)^x+\left(\frac{12}{5}\right)^x\ge2\sqrt{16^x}=2\cdot4^x\)
Cộng theo vế ta có: \(2VT\ge2VP\Leftrightarrow VT\ge VP\)
CMR: moi \(x\in R\) ta có:
\(\left(\frac{12}{5}\right)^x+\left(\frac{15}{4}\right)^x+\left(\frac{20}{3}\right)^x\ge3^x+4^x+5^x\)
Chứng minh rằng với mọi số thực dương thỏa mãn xyz=1
Chứng minh rằng \(\frac{x^3}{\left(1+y\right)\left(1+z\right)}+\frac{y^3}{\left(1+x\right)\left(1+z\right)}+\frac{z^3}{\left(1+x\right)\left(1+y\right)}\ge\frac{3}{4}\)
Chứng minh rằng:
Với x>1; y>0 thì \(\frac{1}{\left(x-1\right)^3}+\left(\frac{x-1}{y}\right)^3+\frac{1}{y^3}\ge3\left(\frac{3-2x}{x-1}+\frac{x}{y}\right)\)
Chứng minh rằng với mọi x > 1 ta luôn có: \(3\left(x^2-\frac{1}{x^2}\right)
Mọi người cho em hỏi:
\(\left(\frac{\sqrt{x}}{\left[\sqrt{x}-3\right]\left[\sqrt{x}+3\right]}+\frac{2}{\sqrt{x}+3}-\frac{3}{\sqrt{x}-3}\right):\left(\sqrt{x}-3+\frac{12-x}{\sqrt{x}+3}\right)\)
\(\left(\frac{\sqrt{x}+2\left[\sqrt{x}-3\right]-3\left[\sqrt{x}+3\right]}{\left[\sqrt{x}-3\right]\left[\sqrt{x}+3\right]}\right):\left(\frac{\left[\sqrt{x}-3\right]\left[\sqrt{x}+3\right]+12-x}{\sqrt{x}-3}\right)\)
\(\left(\frac{\sqrt{x}+2\sqrt{x}-6-3\sqrt{x}-9}{\left[\sqrt{x}+3\right]\left[\sqrt{x}-3\right]}\right):\left(\frac{x+3\sqrt{x}-3\sqrt{x}-9+12-x}{\sqrt{x}+3}\right)\)
\(\left(\frac{-15}{\left[\sqrt{x}+3\right]\left[\sqrt{x}-3\right]}\right):\left(\frac{3}{\sqrt{x}+3}\right)\)
\(\left(\frac{-15}{\left[\sqrt{x}+3\right]\left[\sqrt{x}-3\right]}\right).\left(\frac{\sqrt{x}+3}{3}\right)\)
\(\frac{-5}{\sqrt{x}-3}\)
cho x>1; y>0, chứng minh \(\frac{1}{\left(x-1\right)^3}+\left(\frac{x-1}{y}\right)^3+\frac{1}{y^3}\ge3\left(\frac{3-2x}{x-1}+\frac{x}{y}\right)\)
Cho 3 số thực dương x,y,z có tích bằng 1.Chứng minh rằng:
\(\frac{x^3}{\left(1+y\right)\left(1+z\right)}+\frac{y^3}{\left(1+z\right)\left(1+x\right)}+\frac{z^3}{\left(1+x\right)\left(1+y\right)}\ge\frac{3}{4}\)
Cho 3 số thực dương x,y,z có tích bằng 1.Chứng minh rằng:
\(\frac{x^3}{\left(1+y\right)\left(1+z\right)}+\frac{y^3}{\left(1+z\right)\left(1+x\right)}+\frac{z^3}{\left(1+x\right)\left(1+y\right)}\ge\frac{3}{4}\)
Cho \(\hept{\begin{cases}x,y,z>0\\xy+yz+zx=1\end{cases}}\). Chứng minh rằng:
\(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\ge3+\sqrt{\frac{\left(x+y\right)\left(x+z\right)}{x^2}}+\sqrt{\frac{\left(y+z\right)\left(y+x\right)}{y^2}}+\sqrt{\frac{\left(z+x\right)\left(z+y\right)}{z^2}}\)
