Question

11．已知$f（x）=\left\{\begin{array}{l}2{x^2}-8ax+3，x＜1\\{a^x}-a，x≥1\end{array}\right.$是R上的單調(diào)遞減函數(shù)，則實(shí)數(shù)a的取值范圍為$[{\frac{1}{2}，\frac{5}{8}}]$．

Answer 1

分析 由條件利用函數(shù)的單調(diào)性的性質(zhì)可得$\left\{\begin{array}{l}{2a≥1}\\{0＜a＜1}\\{a-a≤2-8a+3}\end{array}\right.$，由此求得實(shí)數(shù)a的取值范圍．

解答 解：由于已知$f（x）=\left\{\begin{array}{l}2{x^2}-8ax+3，x＜1\\{a^x}-a，x≥1\end{array}\right.$是R上的單調(diào)遞減函數(shù)，故有$\left\{\begin{array}{l}{2a≥1}\\{0＜a＜1}\\{a-a≤2-8a+3}\end{array}\right.$，
求得$\frac{1}{2}$≤a≤$\frac{5}{8}$，
故答案為：$[{\frac{1}{2}，\frac{5}{8}}]$．

點(diǎn)評(píng) 本題主要考查函數(shù)的單調(diào)性的性質(zhì)，體現(xiàn)了轉(zhuǎn)化的數(shù)學(xué)思想，屬于基礎(chǔ)題．