Difference between revisions of "First mean value theorem"
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(Created page with "<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem (First Mean Value Theorem):</strong> Let $f$ and $g$ be bounded and integrable functio...") |
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| − | + | ==Theorem== | |
| − | + | Let $f$ and $g$ be bounded and integrable functions on $[a,b] \cap \mathbb{T}$ and let $g$ be nonnegative (or nonpositive) on $[a,b]\cap\mathbb{T}$. Set | |
$$m=\inf\{f(t) \colon t \in [a,b)\cap\mathbb{T} \}$$ | $$m=\inf\{f(t) \colon t \in [a,b)\cap\mathbb{T} \}$$ | ||
and | and | ||
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Then there exists a number $\Lambda$ satisfying $m \leq \Lambda\leq M$ such that | Then there exists a number $\Lambda$ satisfying $m \leq \Lambda\leq M$ such that | ||
$$\int_a^b f(t)g(t) \Delta t = \Lambda \int_a^b g(t) \Delta t.$$ | $$\int_a^b f(t)g(t) \Delta t = \Lambda \int_a^b g(t) \Delta t.$$ | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 12:43, 16 January 2023
Theorem
Let $f$ and $g$ be bounded and integrable functions on $[a,b] \cap \mathbb{T}$ and let $g$ be nonnegative (or nonpositive) on $[a,b]\cap\mathbb{T}$. Set $$m=\inf\{f(t) \colon t \in [a,b)\cap\mathbb{T} \}$$ and $$M=\sup\{f(t) \colon t \in [a,b)\cap\mathbb{T} \}.$$ Then there exists a number $\Lambda$ satisfying $m \leq \Lambda\leq M$ such that $$\int_a^b f(t)g(t) \Delta t = \Lambda \int_a^b g(t) \Delta t.$$
