已知f(x)对一切x满足xf''(x)+3x[f'(x)]^2=1-e^(-x),若f'(x0)=0,x0≠0,则问f(x0)是不是极值点,如果是,它是极大值还是极小值?

Question

已知f(x)对一切x满足xf''(x)+3x[f'(x)]^2=1-e^(-x),若f'(x0)=0,x0≠0,则问f(x0)是不是极值点,如果是,
它是极大值还是极小值?

Answer 1

xf''(x)+3x[f'(x)]^2=1-e^(-x) (1)
f'(x0)=0代入(1) =>
f''(x) = [1-e^(-x0)] / x0
如果x0>0,e^(-x0) [1-e^(-x0)] >0 => [1-e^(-x0)] / x0>0 => f''(x)>0
如果x01 => [1-e^(-x0)] [1-e^(-x0)] / x0>0 => f''(x)>0
=>f''(x)>0
f'(x0)=0,f''(x)>0 =>f(x0)是极小值