1/\(
\left ( ax+by \right )^{2}\leq \left ( a^{2}+ b^{2} \right ) \left ( x^{2}+ y^{2} \right )
\)
\(
\Leftrightarrow \left ( ax \right )^{2}+2abxy+ \left ( by \right )^{2} \leq \left ( ax \right )^{2} + \left ( ay \right )^{2} + \left ( bx \right )^{2}+ \left ( by \right )^{2}
\)
\(
\Leftrightarrow \left ( ay \right )^{2} + \left ( bx \right )^{2} -2abxy\geq 0
\)
\(
\Leftrightarrow \left ( ay-bx \right )^{2} \geq 0
\),luôn đúng \(
\Rightarrow
\)(ĐPCM)
2/Đặt:
$\begin{cases}\alpha=\left ( \frac{12}{5} \right )^{\frac{x}{2}}>0 \\ \beta=\left ( \frac{15}{4} \right )^{\frac{x}{2}}>0 \\ z=\left ( \frac{20}{3} \right )^{\frac{x}{2}}>0 \end{cases}$
Bất đẳng thức đã cho $\Leftrightarrow \alpha^{2} +\beta^{2}+\gamma^{2}\geq \alpha\beta+\beta\gamma+\gamma\alpha$
$\Leftrightarrow \frac{1}{2}[(\alpha-\beta)^2+(\beta-\gamma)^2+(\gamma-\alpha)^2]\geq 0$ , luôn đúng
$\Rightarrow $ (ĐPCM)