若函数f(x)在[0,1]上连续,在(0,1)内具有二阶导数,f(0)=f(1)=0,f’’(x)0(x∈(0,1));

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问题 若函数f(x)在[0,1]上连续,在(0,1)内具有二阶导数,f(0)=f(1)=0,f’’(x)<0,且f(x)在[0,1]上的最大值为M.求证:
f(x)>0(x∈(0,1));

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答案由题设条件及罗尔定理,[*]∈(0,1),f’(a)=0.由f’’(x)<0(x∈(0,1))=>f’(x)在(0,1)↓ [*] =>f(x)在[0,a]↑,在[a,1]↓ => f(x)>f(0)=0(0<x≤a), f(x)>f(1)=0(a≤x<1), => f(x)>0(x∈(0,1)).

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