设f(x)=xλcos1x, 若x≠00, 若x=0,其导函数在x=0处连续,则λ的取值范围是______
设f(x)=xλcos1x,若x≠00,若x=0,其导函数在x=0处连续,则λ的取值范围是______....
设f(x)=xλcos1x, 若x≠00, 若x=0,其导函数在x=0处连续,则λ的取值范围是______.
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当λ≠0时,有:
①x≠0,f′(x)=λxλ?1cos
+xλ(?sin
)?(?
)=λxλ?1cos
+xλ?2sin
;
②x=0,f′(0)=
=
=
xλ?1cos
当λ=0时,有:
①x≠0,f′(x)=
sin
;
②x=0,f′(0)=
cos
综上所述f′(x)=
由于函数f(x)的导函数在x=0处连续,即
f′(x)=f′(0),即
(λxλ?1cos
+xλ?2sin
①x≠0,f′(x)=λxλ?1cos
1 |
x |
1 |
x |
1 |
x2 |
1 |
x |
1 |
x |
②x=0,f′(0)=
lim |
x→0 |
f(x)?f(0) |
x |
lim |
x→0 |
xλcos
| ||
x |
lim |
x→0 |
1 |
x |
当λ=0时,有:
①x≠0,f′(x)=
1 |
x2 |
1 |
x |
②x=0,f′(0)=
lim |
x→0 |
1 |
x |
1 |
x |
综上所述f′(x)=
|
由于函数f(x)的导函数在x=0处连续,即
lim |
x→0 |
lim |
x→0 |
1 |
x |