Vipul: Created page with "==Definition== Suppose $f$ is a continuous function on a closed interval $[a,b]$ (note that $f$ may be defined on a bigger domain, but w..."

2011-09-04T00:22:46Z

Created page with "==Definition== Suppose <math>f</math> is a continuous function on a closed interval <math>[a,b]</math> (note that <math>f</math> may be defined on a bigger domain, but w..."

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==Definition==

Suppose <math>f</math> is a [[continuous function]] on a [[closed interval]] <math>[a,b]</math> (note that <math>f</math> may be defined on a bigger domain, but we are interested in the restriction of <math>f</math> to the closed interval and require it to be continuous). Then, <math>f</math> attains its minimum and maximum value on the interval. In other words, the following statements are true:

# ''Existence of maximum value'': There is a real number <math>M</math> such that <math>f(x) \le M</math> for all <math>x \in [a,b]</math> and there exists <math>c_1 \in [a,b]</math> such that <math>f(c_1) = M</math>.
# ''Existence of minimum value'': There is a real number <math>m</math> such that <math>f(x) \ge m</math> for all <math>x \in [a,b]</math> and there exists <math>c_2 \in [a,b]</math> such that <math>f(c_2) = m</math>.

==Related facts==

===Applications===

* [[Rolle's theorem]]
* [[Lagrange mean value theorem]]

Extreme value theorem - Revision history

Vipul: Created page with "==Definition== Suppose f is a continuous function on a closed interval [a,b] (note that f may be defined on a bigger domain, but w..."

Vipul: Created page with "==Definition== Suppose $f$ is a continuous function on a closed interval $[a,b]$ (note that $f$ may be defined on a bigger domain, but w..."