(完整版)导数与微分习题及答案

?1?c52．设f?x?满足af?x??bf???，其中a、b、c都是常数，且a?b

?x?x(3) 证明f?x???f??x?

?1?c证：∵af?x??bf????0 ①

?x?x?1?∴af???bf?x??cx ②

?x?①×a-②×b得：a2?b2f?x????ac?bcx x?a?bx?

∴f?x???a?b?x222显然有f??x???f?x?或f?x???f??x? (4) 求f??x?，f???x?

2??ca?bx?c?a???解：f??x??2???b?? 2?222?a?b?x?a?b?x????? f???x???2ac

a2?b2x3??1?2x2,x??1?3??fx??1?x?2， 53．设函数? x, ?12x?16, x?2?(3) 写出f?x?的反函数g?x?的表达式；

?1?x,x??1??2??g?x??? 3x, ?1?x?8

?x?16?, x?812??(4) g?x?是否有间点、不可导点，若有指出这些点。 解：无间断点，但在点x??1和x?8处不可导。

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