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7. 已知曲线 $y=a \mathrm{e}^{x}+x \ln x$ 在点 $(1, a e)$ 处的切线方程为 $y=2 x+b$, 则 ( )

Gaokao · Math · previous-year question

  1. A.$a=e, b=-1$
  2. B.$a=e, b=1$
  3. C.$a=e^{-1}, b=1$
  4. D.$a=e^{-1}, b=-1$correct

Answer

D. $a=e^{-1}, b=-1$

Explanation

【详解】详解: $y^{\prime}=a e^{x}+\ln x+1$, $k=\left.y^{\prime}\right|_{x=1}=a e+1=2$ $\therefore a=e^{-1}$ 将 $(1,1)$ 代人 $y=2 x+b$ 得 $2+b=1, b=-1$, 故选 $\mathrm{D}$.

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