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Math Help - Max and Mins

  1. #1
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    Max and Mins

    Suppose that f:\mathbb{R} \rightarrow \mathbb{R} is continuous on \mathbb{R} and that \lim_{x\to\infty}f = 0.
    How would you prove that f is bounded on R and attains either a max or a min on R?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by frenchguy87 View Post
    Suppose that f:\mathbb{R} \rightarrow \mathbb{R} is continuous on \mathbb{R} and that \lim_{x\to\infty}f = 0.
    How would you prove that f is bounded on R and attains either a max or a min on R?
    It doesn't have to. f(x)=e^{-x}
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    It doesn't have to. f(x)=e^{-x}

    Perhaps he meant the limit as x approaches both + and - sends f(x) to 0?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by southprkfan1 View Post
    Perhaps he meant the limit as x approaches both + and - sends f(x) to 0?
    If so, I will help with the bounded part. The other is up to you. Since \lim_{x\to\infty}f(x)=\lim_{x\to-\infty}f(x)=0 there exists some T>0 such that x\in(-\infty,-T)\cup(T,\infty) implies that |f(x)|<1. But, then we have that f:[-T,T]\mapsto\mathbb{R} is bounded (since [-T,T] is compact and f continuous). Thus, \left|f(x)\right|\leqslant M,\text{ }x\in[-T,T] for some M\in\mathbb{R}. It readily follows that |f(x)|\leqslant\max\left\{1,M\right\} for all x\in\mathbb{R}
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