※ 引述《WinVNC (。)》之銘言：
: 在練習卷上看到的一題，Hint是f incresing <=> f' >= 0
: 實在想不出來該怎麼做，還請高手們指點
: Let f be continuous on [0,1] , f(0)=0
: f'(x) finite for every x in (0,1).
: Prove that if f' is an increasing function on (0,1),
: then so is the function g defined by the equation g(x)= f(x)/x
Observe that: g is differentiable and g'(x) = (xf'(x)-f(x))/x^2.
g is increasing if and only if xf'(x)-f(x)≧0 for any x sits in (0,1).
This holds if and only if f'(x)≧f(x)/x = (f(x)-f(0))/(x-0) = f'(y) by MVT.
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