Question

若函數(shù)e=kx²-4x+k-3對一切實(shí)數(shù)x都有e＜它，則實(shí)數(shù)k的取值范圍是______．

Answer 1

∵函數(shù)y=kx²-4x+k-3對一切實(shí)數(shù)x都有y＜p
∴函數(shù)y=kx²-4x+k-3圖圖象全部在x軸圖下方
①當(dāng)k=p時函數(shù)y=-4x-3顯然此時函數(shù)圖圖象不全部在x軸圖下方
所以k=p不符合題意
②當(dāng)k≠p時原函數(shù)是二次函數(shù)
∵函數(shù)y=kx²-4x+k-3對一切實(shí)數(shù)x都有y＜p
∴二次函數(shù)y=kx²-4x+k-3圖圖象全部在x軸圖下方
所以

k＜p △=16-4k(k-3)＜p

解得k＜-1
由①②可得實(shí)數(shù)k圖取值范圍是 （-∞，-1）．
故答案為：（-∞，-1）．