高数题:设f(x)在R上有二阶连续导数,且f(0)=0,x不等于0时,g(x)=f(x)/x;x=0时,g(x)=f'(0)证g'(x)在R上有一阶连续导数.下面好像是个提示:x不等于0时,g'(x)=(xf'(x)-f(x))/x^2,x等于0时,g'(x)=1/2f'(0) 时间很紧迫,
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![高数题:设f(x)在R上有二阶连续导数,且f(0)=0,x不等于0时,g(x)=f(x)/x;x=0时,g(x)=f'(0)证g'(x)在R上有一阶连续导数.下面好像是个提示:x不等于0时,g'(x)=(xf'(x)-f(x))/x^2,x等于0时,g'(x)=1/2f'(0) 时间很紧迫,](/uploads/image/z/679688-8-8.jpg?t=%E9%AB%98%E6%95%B0%E9%A2%98%EF%BC%9A%E8%AE%BEf%28x%29%E5%9C%A8R%E4%B8%8A%E6%9C%89%E4%BA%8C%E9%98%B6%E8%BF%9E%E7%BB%AD%E5%AF%BC%E6%95%B0%2C%E4%B8%94f%280%29%3D0%2Cx%E4%B8%8D%E7%AD%89%E4%BA%8E0%E6%97%B6%2Cg%28x%29%3Df%28x%29%2Fx%EF%BC%9Bx%3D0%E6%97%B6%2Cg%28x%29%3Df%27%280%29%E8%AF%81g%27%28x%29%E5%9C%A8R%E4%B8%8A%E6%9C%89%E4%B8%80%E9%98%B6%E8%BF%9E%E7%BB%AD%E5%AF%BC%E6%95%B0.%E4%B8%8B%E9%9D%A2%E5%A5%BD%E5%83%8F%E6%98%AF%E4%B8%AA%E6%8F%90%E7%A4%BA%EF%BC%9Ax%E4%B8%8D%E7%AD%89%E4%BA%8E0%E6%97%B6%2Cg%27%28x%29%3D%28xf%27%28x%29-f%28x%29%29%2Fx%5E2%2Cx%E7%AD%89%E4%BA%8E0%E6%97%B6%2Cg%27%28x%29%3D1%2F2f%27%280%29+%E6%97%B6%E9%97%B4%E5%BE%88%E7%B4%A7%E8%BF%AB%2C)
高数题:设f(x)在R上有二阶连续导数,且f(0)=0,x不等于0时,g(x)=f(x)/x;x=0时,g(x)=f'(0)证g'(x)在R上有一阶连续导数.下面好像是个提示:x不等于0时,g'(x)=(xf'(x)-f(x))/x^2,x等于0时,g'(x)=1/2f'(0) 时间很紧迫,
高数题:设f(x)在R上有二阶连续导数,且f(0)=0,x不等于0时,g(x)=f(x)/x;x=0时,g(x)=f'(0)
证g'(x)在R上有一阶连续导数.下面好像是个提示:x不等于0时,g'(x)=(xf'(x)-f(x))/x^2,x等于0时,g'(x)=1/2f'(0) 时间很紧迫,
高数题:设f(x)在R上有二阶连续导数,且f(0)=0,x不等于0时,g(x)=f(x)/x;x=0时,g(x)=f'(0)证g'(x)在R上有一阶连续导数.下面好像是个提示:x不等于0时,g'(x)=(xf'(x)-f(x))/x^2,x等于0时,g'(x)=1/2f'(0) 时间很紧迫,
应该是证g(x)在R上有一阶连续导数吧?
当x≠0时,g(x)=f(x)/x
∴g'(x) = [xf'(x)-f(x)]/x²
g'(x)在x≠0时连续
x=0时,
g'(0) = lim(x→0) [g(x)-g(0)]/(x-0)
=lim(x→0) [f(x)/x-f'(0)]/x
=lim(x→0) [f(x)-xf'(0)]/x²
=lim(x→0) [f'(x)-f'(0)]/(2x)
=(1/2)f''(0)
又lim(x→0) [xf'(x)-f(x)]/x²
=lim(x→0) [f'(x)+xf''(x)-f'(x)]/(2x)
=(1/2)f''(0)
∴lim(x→0) g'(x) =g'(0)
即g'(x)在x=0处连续
综上可得g'(x)在R上连续,即g(x)在R上有一阶连续导数
证明:x不等于0时,g'(x)=(xf'(x)-f(x))/x^2,
x等于0时,g'(0)=lim(g(x)-g(0))/x=lim(f(x)/x-f'(0))/x
=lim(f(x)-xf'(0))/x^2=lim(f'(x)-f'(0))/2x=1/2f''(0)
x趋于0时,limg'(x)=(xf'(x)-f(x))/x^2,=lim(f'(x)+xf''(x)-f('x))/2x=limf''(x)/2=f''(0)/2 =g'(0)
所以:g'(x)在R上连续
应该是证g(x)在R上有一阶连续导数吧?加油 你是最棒的
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