Let f be a function, defined and continuous on the entire real line, with the property...

 

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Let f be a function, defined and continuous on the entire real line, with the property that for every real number C, there is another real number D such that |x ≥D = f(x) > C. (An informal way to say this is: f(x)→∞ as x→ ±∞o. For example, the function f(x) = r² has this property- given C, we may take D to C- but the function g(x) = ³ does not.) be Apply the Extreme Value Theorem to show that f is bounded below, and attains its minimum: that is, that there is a real number 3 such that f(3) ≤ f(x) for every real number z. Let f be a function, defined and continuous on the entire real line, with the property that for every real number C, there is another real number D such that |x ≥D = f(x) > C. (An informal way to say this is: f(x)→∞ as x→ ±∞o. For example, the function f(x) = r² has this property- given C, we may take D to C- but the function g(x) = ³ does not.) be Apply the Extreme Value Theorem to show that f is bounded below, and attains its minimum: that is, that there is a real number 3 such that f(3) ≤ f(x) for every real number z.