User:Egm6341.s10.team4.anandankala/HW1-4

=HW Problem 1.4=

(a) Prove the Integral Mean Value Theorem (pg.2-3)for the case where W(x) is non negative,i.e W(x)$$\geq$$0.

(b) Prove also another version of the integral mean value theorem; W(x)$$\neq$$0, $$\forall$$ x $$\in$$[a,b].

=Solution=

According to Integral Mean Value Theorem, $$\displaystyle\int^b_a W(x)f(x)\,dx$$=$$\displaystyle f(\xi)\int^b_a W(x)\,dx$$, where W(x)$$\geq$$0, $$\forall$$ x $$\in$$[a,b].

Let f(x) be a function continuous on [a,b], which implies that there exists a minimum m and a maximum M such that

$$m\leq f(x)\leq M$$,

$$\forall$$ x $$\in$$[a,b], where m and M are constants.

Multiplying with $$\displaystyle\int^b_a W(x)\,dx$$ on all the sides. we get

$$m \displaystyle\int^b_a W(x)\,dx$$$$\leq$$ $$f(x) \displaystyle\int^b_a W(x)\,dx$$$$\leq$$$$ M \displaystyle\int^b_a W(x)\,dx$$.

$$\Rightarrow$$ $$m \displaystyle\int^b_a W(x)\,dx$$$$\leq$$$$\displaystyle\int^b_a f(x)W(x)\,dx$$$$\leq$$$$ M \displaystyle\int^b_a W(x)\,dx$$.

$$\Rightarrow$$ $$m \leq$$  $$\dfrac{\displaystyle\int^b_a W(x)f(x)\,dx}{\displaystyle\int^b_a W(x)\,dx}$$  $$\leq M$$ $$\exists$$     $$\xi \in$$ [a,b] such that $$f(\xi)$$ exists between m and M.

$$\Rightarrow$$  $$f(\xi)= \dfrac{\displaystyle\int^b_a W(x)f(x)\,dx}{\displaystyle\int^b_a W(x)\,dx}$$.

$$\Rightarrow$$ $$\displaystyle\int^b_a W(x)f(x)\,dx$$= $$f(\xi) \displaystyle\int^b_a W(x)\,dx$$.

Hence the Integral mean value theorem is proved for W(x)$$\geq$$0

Consider the case when W(x)$$\leq$$0.

we have m $$\leq$$ f(x) $$\leq$$ M ,

Multiplying with $$\displaystyle\int^b_a W(x)\,dx$$ on all the sides and since W(x)$$\leq$$0. we get

M $$\displaystyle\int^b_a W(x)\,dx$$$$\leq$$$$\displaystyle\int^b_a f(x)W(x)\,dx$$$$\leq$$ m $$\displaystyle\int^b_a W(x)\,dx$$.

$$\Rightarrow$$ M $$\geq$$  $$\dfrac{\displaystyle\int^b_a W(x)f(x)\,dx}{\displaystyle\int^b_a W(x)\,dx}$$  $$\geq$$ m

$$\exists$$    ζ $$\in$$ [a,b] such that f(ζ) exists between m and M.

$$\Rightarrow$$  $$f(\zeta)=\dfrac{\displaystyle\int^b_a W(x)f(x)\,dx}{\displaystyle\int^b_a W(x)\,dx}$$.

$$\Rightarrow$$ $$\displaystyle\int^b_a W(x)f(x)\,dx$$= $$f(\zeta) \displaystyle\int^b_a W(x)\,dx$$.

Hence the Integral Mean Value Theorem is proved for the case when W(x)$$\leq$$0.