Question

已知函數(shù)=,=,若曲線和曲線都過(guò)點(diǎn)P(0,2),且在點(diǎn)P處有相同的切線

(Ⅰ)求,,,的值;

(Ⅱ)若≥-2時(shí),≤,求的取值范圍.

Answer 1

 (Ⅰ)由已知得,

而=,=,∴=4,=2,=2,=2;  

(Ⅱ)由(Ⅰ)知,,,

設(shè)函數(shù)==(),

==,

有題設(shè)可得≥0,即,

令=0得,=,=-2,

(1)若,則-2<≤0,∴當(dāng)時(shí),<0,當(dāng)時(shí),>0,即在單調(diào)遞減,在單調(diào)遞增,故在=取最小值,而==≥0,

∴當(dāng)≥-2時(shí),≥0,即≤恒成立,

(2)若,則=,

∴當(dāng)≥-2時(shí),≥0,∴在(-2,+∞)單調(diào)遞增,而=0,

∴當(dāng)≥-2時(shí),≥0,即≤恒成立,

(3)若,則==<0,

∴當(dāng)≥-2時(shí),≤不可能恒成立,

綜上所述,的取值范圍為[1,].