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[几何] 2020年天津卷第20题 导数之双变量

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isee posted 2020-7-17 22:55 |Read mode
已知函数$f(x)=x^3+k\ln x(k\in \mathrm R)$,$f'(x)$为$f(x)$的导函数.
(Ⅰ)当$k=6$时,
      (i)求曲线$y=f(x)$在点$(1,f(1))$处的切线方程;
      (ii)求函数$g(x)=f(x)-f'(x)+\frac9x$的单调区间和极值;
(Ⅱ)当$k\ge -3$时,求证:对任意的$x_1,x_2\in [1,+\infty )$,且$x_1>x_2$,有$\frac {f'\left( x_1 \right)+f'\left( x_2 \right)}2>\frac{f\left( x_1 \right)-f\left( x_2 \right)}{x_1-x_2}$.

2020年天津卷第20题


感觉本次导数倒是回归到常规了,天津卷

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