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What theorem is this?

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
3 years, 10 months 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. · 3 years, 10 months ago