共50条信息
已知正数\(x\),\(y\),\(z\)满足\(x^{2}+y^{2}+z^{2}=6\).
\((\)Ⅰ\()\)求\(x+2y+z\)的最大值;
\((\)Ⅱ\()\)若不等式\(|a+1|-2a\geqslant x+2y+z\)对满足条件的\(x\),\(y\),\(z\)恒成立,求实数\(a\)的取值范围.
实数\(x\),\(y\),\(z\)满足\(x+2y+3z=a(a\)为常数\()\),则\(x^{2}+y^{2}+z^{2}\)的最小值为\((\) \()\)
已知\(a\),\(b∈(0,+∞)\),\(a+b=1\),\(x_{1}\),\(x_{2}∈(0,+∞)\).
\((1)\)求\(\dfrac{{{x}_{1}}}{a}+\dfrac{{{x}_{2}}}{b}+\dfrac{2}{{{x}_{1}}{{x}_{2}}}\)的最小值;
\((2)\)求证:\((ax_{1}+bx_{2})(ax_{2}+bx_{1})\geqslant x_{1}x_{2}\).
若实数\(x+y+z=1\),则\(2x^{2}+y^{2}+3z^{2}\) 的最小值为\((\) \()\)
已知\(a{,}b{,}c{∈}(0{,}1)\),且\({ab}{+}{bc}{+}{ac}{=}1\),则\(\dfrac{1}{1{-}a}{+}\dfrac{1}{1{-}b}{+}\dfrac{1}{1{-}c}\)的最小值为\(({ })\)
三个实数\(x\),\(y\),\(z\)满足\(2x+3y+z=13\),\(4x^{2}+9y^{2}+z^{2}-2x+15y+3z=82\),则\(xyz=\)________.
已知\(x\),\(y\),\(z∈R\),且\(\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{3}{z}=1\),则\(x+\dfrac{y}{2}+\dfrac{z}{3}\)的最小值是\((\) \()\)
\([\)选修\(4-5\):不等式选讲\(]\)
已知\(a\),\(b\),\(c\),\(d\)均为正数,且\(ad=bc\).
\((\)Ⅰ\()\)证明:若\(a+d > b+c\),则\(|a-d| > |b-c|\);
\((\)Ⅱ\()\)若\(t· \sqrt{a^{2}+b^{2}}· \sqrt{c^{2}+d^{2}}= \sqrt{a^{4}+c^{4}}+ \sqrt{b^{4}+d^{4}}\),求实数\(t\)的取值范围.
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