作业帮当α,β取何值时,函数f(x)={(x^α cos〖1/x〗 ,x>0;β+sin〖x^2 〗 ,x≤0)┤在x=0处连
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当α,β取何值时,函数f(x)={(x^α cos〖1/x〗 ,x>0;β+sin〖x^2 〗 ,x≤0)┤在x=0处连续:何时在x=0处可导
f(x) = x^α cos (1/x) ; x >0
= β+sin(x^2) ; x≤0
f(0+) = lim(x->0+){ x^α cos (1/x) }
=0
f(0-) =lim(x->0-){ β+sin(x^2) }
= β
f is cts on x = 0 ,=> β =0
f(x) = x^α cos (1/x) ; x >0
= sin(x^2) ; x≤0
x >0
f'(x) = αx^(α-1) cos (1/x) + x^(α-2) sin(1/x)
f'(0+) = 0
x≤0
f'(x) = -2xcos(x^2)
f'(0-) =0
f 在x=0处可导
= β+sin(x^2) ; x≤0
f(0+) = lim(x->0+){ x^α cos (1/x) }
=0
f(0-) =lim(x->0-){ β+sin(x^2) }
= β
f is cts on x = 0 ,=> β =0
f(x) = x^α cos (1/x) ; x >0
= sin(x^2) ; x≤0
x >0
f'(x) = αx^(α-1) cos (1/x) + x^(α-2) sin(1/x)
f'(0+) = 0
x≤0
f'(x) = -2xcos(x^2)
f'(0-) =0
f 在x=0处可导
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