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Given \(f\) is continous function at every real numbers, prove that \(f\) does have global maxima if given condition:

\[\lim_{x \to \infty} f(x) = \lim_{x \to -\infty} f(x) = -\infty\]

Note by Nabila Nida Rafida
4 years, 1 month ago

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Use Rolle's Theorem(Or Lagrange's Mean Value Theorem as a general case) ..http://en.wikipedia.org/wiki/Rolle's_theorem..

Using it we see that \(f'(c)=0\) for some real \(c\).

We cannot have a global minima for \(f\) as \(-\infty\) is the least "value" that any function can "attain". So \(f\) has at least one global maximum.

E.g.

\(\rightarrow\)Look at the graph of \(y=-x^{2}\).. It satisfies the given conditions and has a global maxima at x=0.

Krishna Jha - 4 years, 1 month ago

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