Question

15．若命題：“?x∈R，kx²-kx-1≥0”是假命題，則實數k的取值范圍是（-4，0]．

Answer 1

分析 若命題：“?x∈R，kx²-kx-1≥0”是假命題，則命題：“?x∈R，kx²-kx-1＜0”是真命題，則k=0，或$\left\{\begin{array}{l}k＜0\\{k}^{2}+4k＜0\end{array}\right.$，解得答案．

解答 解：若命題：“?x∈R，kx²-kx-1≥0”是假命題，
則命題：“?x∈R，kx²-kx-1＜0”是真命題，
∴k=0，或$\left\{\begin{array}{l}k＜0\\{k}^{2}+4k＜0\end{array}\right.$，
解得：k∈（-4，0]，
故答案為：（-4，0]

點評 本題以命題的真假判斷與應用為載體，考查了特稱命題，二次函數的圖象和性質，難度中檔．