设f(x)在[a,b]上满足|f’’(x)|≤2,且f(x)在(a,b)内取到最小值。证明:|f’(a)|+|f’(b)|≤2(b—a).

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问题 设f(x)在[a,b]上满足|f’’(x)|≤2,且f(x)在(a,b)内取到最小值。证明:|f(a)|+|f(b)|≤2(b—a).

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答案因为f(x)在(a,b)内取到最小值,所以存在c∈(a,b),使得f(c)为f(x)在[a,b]上的最小值,从而f(c)=0. [*]

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