Question

已知函數(shù)=,=,若曲線和曲線都過點(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); (Ⅱ)的取值范圍為[1,].

解析試題分析：(Ⅰ)先由過點(diǎn)得出，再求在點(diǎn)導(dǎo)數(shù)，由導(dǎo)數(shù)幾何意義知，從而解得；
(Ⅱ)設(shè)==()=, 由題設(shè)可得≥0,即, 令=0得,=,="-2," 對(duì)分3中情況討論得出結(jié)果.
試題解析：(Ⅰ)由已知得,
而=,=,∴=4,=2,=2,="2;"  
(Ⅱ)由(Ⅰ)知,,, 設(shè)函數(shù)
==(),==, 由題設(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),