问答题
设函数f(x)和g(x)都可导，且F(x)=g(x)|f(x)|，求证：
问答题
当f(x
0
)≠0时，F(x)在点x=x
0
处必可导；
【正确答案】
【答案解析】当f(x
0
)≠0时，由f(x)的连续性知：存在δ＞0，使得当|x-x
0
|＜δ时f(x)与f(x
0
)同号．若f(x
0
)＞0，则当|x-x
0
|＜δ时有
F(x)=g(x)|f(x)|=g(x)f(x)，
从而F(x)在点x=x 0 处可导(且F"(x 0 )=g(x 0 )f"(x 0 )+g"(x 0 )f(x 0 )，类似可证当f(x 0 )＜0时，F(x)也在x=x 0 处可导)．
F(x)=g(x)|f(x)|=g(x)f(x)，
从而F(x)在点x=x 0 处可导(且F"(x 0 )=g(x 0 )f"(x 0 )+g"(x 0 )f(x 0 )，类似可证当f(x 0 )＜0时，F(x)也在x=x 0 处可导)．
问答题
当f(x
0
)=0时，F(x)在点x=x
0
处可导的充分必要条件是f"(x
0
)g(x
0
)=0．
【正确答案】
【答案解析】当f(x
0
)=0时，F(x
0
)=g(x
0
)|f(x
0
)|=0，从而F(x)在点x=x
0
处的左导数与右导数分别是
故当f(x 0 )=0时，|F(x)|在点x=x 0 处可导的充分必要条件为
F" - (x 0 )=F" + (x 0 )，即-g(x 0 )|f"(x 0 )|=g(x 0 )|f"(x 0 )|，
得
故当f(x 0 )=0时，|F(x)|在点x=x 0 处可导的充分必要条件为
F" - (x 0 )=F" + (x 0 )，即-g(x 0 )|f"(x 0 )|=g(x 0 )|f"(x 0 )|，
得