Question

设f(x)二阶可导,且f``(x)>0,f(0)=0,则F(x)=f(x)/x在(0,正无穷)内单调递增还是递减?

Answer 1

F‘(x)=[xf'(x)-f(x)]/x²

令 g(x)=xf'(x)-f(x),x≥0,则 g'(x)=f'(x)+xf''(x)-f'(x)=xf''(x)≥0

从而 g(x)在[0,+∞)是增,当x>0时,有g(x)>g(0)=0

从而 F'(x)=g(x)/x²>0,F(x)在 (0,+∞)是增