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            • 1.

              已知\(f\left( \left. x+ \dfrac{1}{x} \right. \right)=x^{2}+ \dfrac{1}{x^{2}}\),则\(f(x)\)的解析式为________.

            • 2.

              设二次函数\(f(x)=ax^{2}+(2b+1)x-a-2(a,b∈R,a\neq 0)\)在\([3,4]\)上至少有一个零点,则\(a^{2}+b^{2}\)的最小值为\((\)    \()\)

              A.\(\dfrac{1}{100}\)
              B.\(\dfrac{1}{10}\)
              C.\(\dfrac{4}{289}\)
              D.\(\dfrac{1}{{{(2\sqrt{5}+4)}^{2}}}\)
            • 3. 如图,在正方形\(ABCD\)中,\(AB=2\),点\(E\),\(F\)分别在边\(AB\),\(DC\)上,\(M\)为\(AD\)的中点,且\(\overrightarrow{ME}· \overrightarrow{MF}=0 \)\(∆MEF \)的面积的取值范围为      \((\)  \()\)

              A.\(\left[1, \dfrac{5}{4}\right] \)
              B.\(\left[1,2\right] \)
              C.\(\left[ \dfrac{1}{2}, \dfrac{5}{4}\right] \)
              D.\(\left[ \dfrac{1}{2}, \dfrac{3}{2}\right] \)
            • 4.

              若不等式\(x^{2}+ax+1\geqslant 0\)对一切\(x∈\left( \left. 0, \dfrac{1}{2} \right. \right]\)恒成立,则\(a\)的最小值为\((\)  \()\)

              A.\(0\)
              B.\(-2\)
              C.\(- \dfrac{5}{2}\)
              D.\(-3\)
            • 5.

              \((1)\)已知\(x < -2\),求函数\(y=2x+ \dfrac{1}{x+2}\)的最大值;

              \((2)\)求\(y= \dfrac{x^{2}+5}{ \sqrt{x^{2}+4}}\)的最小值;

              \((3)\)若正数\(a\),\(b\)满足\(ab=a+b+3\),求\(a+b\)的取值范围.

            • 6.

              若不等式\(x^{2}+ax+1\geqslant 0\)对于一切\(x∈(0, \dfrac{1}{2} ]\)恒成立,则\(a\)的最小值是(    )

              A.\(0\)  
              B.\(-2\)  
              C.\(- \dfrac{5}{2} \)
              D.\(-3\)
            • 7.

              探究函数\(f(x){=}2x{+}\dfrac{8}{x}{,}x{∈}(0{,} + \infty)\)的最小值,并确定取得最小值时\(x\)的值\(.\)列表如下:

              \(x\)

              \(…\)

              \(0.5\)

              \(1\)

              \(1.5\)

              \(1.7\)

              \(1.9\)

              \(2\)

              \(2.1\)

              \(2.2\)

              \(2.3\)

              \(3\)

              \(4\)

              \(5\)

              \(7\)

              \(…\)

              \(y\)

              \(…\)

              \(16\)

              \(10\)

              \(8.34\)

              \(8.1\)

              \(8.01\)

              \(8\)

              \(8.01\)

              \(8.04\)

              \(8.08\)

              \(8.6\)

              \(10\)

              \(11.6\)

              \(15.14\)

              \(…\)

              请观察表中\(y\)值随\(x\)值变化的特点,完成以下的问题.
              \((1)\)函数\(f(x){=}2x{+}\dfrac{8}{x}(x{ > }0)\)在区间\((0,2)\)上递减;函数\(f(x){=}2x{+}\dfrac{8}{x}(x{ > }0)\)在区间 ______上递增\(.\)当\(x= \)______时,\(y_{最小}= \)______.
              \((2)\)证明:函数\(f(x){=}2x{+}\dfrac{8}{x}(x{ > }0)\)在区间\((0,2)\)递减.
              \((3)\)思考:函数\(y=2x+\dfrac{8}{x}(x > 0)\)时,有最值吗?是最大值还是最小值?此时\(x\)为何值?\((\)直接回答结果,不需证明\()\)
            • 8. 已知函数\(f(x)=x+ \dfrac {t}{x}(t > 0)\)有如下性质:该函数在\((0, \sqrt {t}]\)上是减函数,在\([ \sqrt {t},+∞)\)是增函数
              \((1)\)若\(g(x+ \dfrac {1}{x})=x^{2}+ \dfrac {1}{x^{2}}\),求\(g(x)\)的解析式
              \((2)\)已知函数\(h(x)= \dfrac {4x^{2}-12x-3}{2x+1}(x∈[0,1])\),利用上述性质,求\(h(x)\)的值域.
            • 9.

              已知函数\(f(x)=x+\dfrac{4}{x}\),\(g(x)={{2}^{x}}+a\),若\(\forall {{x}_{1}}\in \left[ \dfrac{1}{2},1 \right],\exists {{x}_{2}}\in \left[ 2,3 \right],\)使得\(f\left( {{x}_{1}} \right)\geqslant g\left( {{x}_{2}} \right)\),则实数\(a\)的取值范围是________.

            • 10.

              已知函数\(f(x)=\begin{cases}\left|{\log }_{2}\left(-x\right)\right|,x < 0 \\ {x}^{2}-2x+2,x\geqslant 0\end{cases} \),函数\(F(x)=f(x)-a\)有四个不同的零点\(x_{1}\),\(x_{2}\),\(x_{3}\),\(x_{4}\)且满足:\(x_{1} < x_{2} < x_{3} < x_{4}\),则\(\dfrac{{{x}_{{2}}}}{{{x}_{{1}}}}+\dfrac{{{x}_{{3}}}x_{{1}}^{{2}}+{{x}_{{4}}}x_{{1}}^{{2}}}{{2}}\)的取值范围为________.

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