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==
<strong>Theorem (First Mean Value Theorem):</strong> 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
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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
 
$$\int_a^b f(t)g(t) \Delta t = \Lambda \int_a^b g(t) \Delta t.$$
 
$$\int_a^b f(t)g(t) \Delta t = \Lambda \int_a^b g(t) \Delta t.$$
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<strong>Proof:</strong> █
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==Proof==
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==References==
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[[Category:Theorem]]
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[[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 $$\int_a^b f(t)g(t) \Delta t = \Lambda \int_a^b g(t) \Delta t.$$

Proof

References