Question

20．已知函數(shù)f（x）=$\left\{\begin{array}{l}{1-{e}^{-x}，x≤0}\\{\sqrt{2x}，x＞0}\end{array}\right.$，若|f（x）|≥ax，則實(shí)數(shù)a的取值范圍為（　�。�

• A． （-∞，0]
• B． （-∞，-1]
• C． [-2，0]
• D． [-1，0]

Answer 1

分析 畫出函數(shù)y=|f（x）|=$\left\{\begin{array}{l}{（\frac{1}{e}）}^{x}-1，x≤0\\ \sqrt{2x}，x＞0\end{array}\right.$的圖象，數(shù)形分析可得實(shí)數(shù)a的取值范圍為[y′|_x=0，0]，求導(dǎo)可得答案．

解答 解：∵f（x）=$\left\{\begin{array}{l}1-{e}^{-x}，x≤0\\ \sqrt{2x}，x＞0\end{array}\right.$，
∴函數(shù)y=|f（x）|=$\left\{\begin{array}{l}{（\frac{1}{e}）}^{x}-1，x≤0\\ \sqrt{2x}，x＞0\end{array}\right.$的圖象如下圖所示：

∵y′=$\left\{\begin{array}{l}-{（\frac{1}{e}）}^{x}，x≤0\\ \frac{\sqrt{2x}}{2x}，x＞0\end{array}\right.$，
故y′|_x=0=-1，
故若|f（x）|≥ax，則a∈[-1，0]，
故選：D

點(diǎn)評(píng) 本題考查的知識(shí)點(diǎn)是分段函數(shù)的應(yīng)用，恒成立問題，導(dǎo)數(shù)的幾何意義，難度中檔．