Mean value theorem

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Theorem (First Mean Value 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: