如果函数y=f(x)在x=0处得导数存在,且f(x)=f(-x)求f‘(0)的值
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如果函数y=f(x)在x=0处得导数存在,且f(x)=f(-x)求f‘(0)的值
![如果函数y=f(x)在x=0处得导数存在,且f(x)=f(-x)求f‘(0)的值](/uploads/image/z/16812740-20-0.jpg?t=%E5%A6%82%E6%9E%9C%E5%87%BD%E6%95%B0y%3Df%28x%29%E5%9C%A8x%3D0%E5%A4%84%E5%BE%97%E5%AF%BC%E6%95%B0%E5%AD%98%E5%9C%A8%2C%E4%B8%94f%EF%BC%88x%EF%BC%89%3Df%EF%BC%88-x%EF%BC%89%E6%B1%82f%E2%80%98%EF%BC%880%EF%BC%89%E7%9A%84%E5%80%BC)
f(x) = f(-x)
f'(0-)= lim(y->0-) { [f(y) - f(0)]/y }
= lim(y->0-) { [f(-y) - f(0)] / (y) }
= -lim(y-> 0-) { [f(-y) - f(0)]/ (-y) }
= -lim(-y-> 0+){ [f(-y) - f(0)]/ (-y) }
= - f'(0+)
f'(0) => f'(0+) = f'(0-)
=> f'(0)= 0
f'(0-)= lim(y->0-) { [f(y) - f(0)]/y }
= lim(y->0-) { [f(-y) - f(0)] / (y) }
= -lim(y-> 0-) { [f(-y) - f(0)]/ (-y) }
= -lim(-y-> 0+){ [f(-y) - f(0)]/ (-y) }
= - f'(0+)
f'(0) => f'(0+) = f'(0-)
=> f'(0)= 0
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