名 匿 asked in 科學數學 · 9 years ago

高中數學證明兩題

證明題:

證明x^5=2011有正實根。

證明x^5=2011恰有一正實根。

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  • 9 years ago
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    1. existence(存在正實根)

    f(x)=x^5 is continuous function.(連續函數)

    f(0) <2011, f(5) > 2011, so there exists k in (0,5), f(k)=2011

    (Note: k is positive number(正數).)

    2. uniqueness(唯一)

    f(x)=x^5 is increasing function. (漸增函數)

    If a < b and a, b both are solutions of x^5=2011 then a^5=b^5=2011.

    But a^5 < b^5 (since x^5 is increasing) contradicts to a^5=b^5,

    so that it is impossible to find two different solutions of x^5=2011.

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