Question

19．已知不等式kx²-2x+6k＜0 （k≠0），若不等式的解集為∅，則k的取值范圍為k≥$\frac{\sqrt{6}}{6}$．

Answer 1

分析 根據(jù)題意，得出不等式組$\left\{\begin{array}{l}{k＞0}\\{△≤0}\end{array}\right.$，求出解集即可．

解答 解：∵不等式kx²-2x+6k＜0 （k≠0）的解集為∅，
∴應(yīng)滿足$\left\{\begin{array}{l}{k＞0}\\{△≤0}\end{array}\right.$，
即$\left\{\begin{array}{l}{k＞0}\\{4-2{4k}^{2}≤0}\end{array}\right.$；
解得$\left\{\begin{array}{l}{k＞0}\\{k≥\frac{\sqrt{6}}{6}或k≤-\frac{\sqrt{6}}{6}}\end{array}\right.$，
∴k的取值范圍是k≥$\frac{\sqrt{6}}{6}$．
故答案為：k≥$\frac{\sqrt{6}}{6}$．

點(diǎn)評(píng) 本題考查一元二次不等式與二次函數(shù)的應(yīng)用問(wèn)題，也考查了判別式的應(yīng)用問(wèn)題，是基礎(chǔ)題目．