What theorem is this?

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

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

#Calculus #Space #Math

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Comments

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 - 7 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$