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设f(x)在[0 +∞)上可微 且0≤f(x)≤f(x) f(0)=0。证明:在[0 +∞)上f(x)≡0。

设f(x)在[0,+∞)上可微,且0≤f(x)≤f(x),f(0)=0。证明:在[0,+∞)上f(x)≡0。

请帮忙给出正确答案和分析,谢谢!

发布时间:2020-08-21
参考答案

正确答案:令g(x)=e-xf(x)x∈[0+∞)则 g(0)=f(0)=0g(x)≥0x∈(0+∞) ① 由g"(x)=e-x(f"(x)-f(x))≤0x∈[0+∞)因此g(x)为[0+∞)的单调递减函数 g(x)≤g(0)=0x∈[0+∞) ② 由①②可知 g(x)≡0=>f(x)≡0x∈[0+∞)。
令g(x)=e-xf(x),x∈[0,+∞),则g(0)=f(0)=0,g(x)≥0,x∈(0,+∞)①由g"(x)=e-x(f"(x)-f(x))≤0,x∈[0,+∞),因此g(x)为[0,+∞)的单调递减函数,g(x)≤g(0)=0,x∈[0,+∞)②由①②可知g(x)≡0=>f(x)≡0,x∈[0,+∞)。

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