FS17

5

Seien $F:[a,b]\to \mathbb{R}$, $G:[a,b]\to \mathbb{R}$ integrierbare Funktionen mit $G$ stetig und $F>0$. Zeigen sie dass $c\in [a,b]$ existiert, so dass

$${\int }_a^{b}F(x)G(x)dx=G(x){\int }_a^{b}F(x)dx$$

Proof: Define new function, show that $G$ takes it’s max and min, use ZWS.

• since $G$ ist continuous on the compact interval $[a,b]$ it attains it’s extrema. There are $x_m,x_M\in [a,b]$ with

$$m:=G(x_m)=\underset{x\in [a,b]}{\min }G(x)M:=G(x_M)=\underset{x\in [a,b]}{\max }G(x)$$

• FG is integrable since it’s a product of integrable functions.
• Since $F>0$

$$mF(x)\le F(x)G(x)\le MF(x)$$

• and by monotonicity of the integral

$$m{\int }_a^{b}F\le {\int }_a^{b}FG\le M{\int }_a^{b}F$$

• Define $H:[a,b]\to \mathbb{R}$, $H(x):=G(x){\int }_a^{b}F(t)dt$. As ${\int }_a^{b}F$ is a constant and $G$ is continous, Note $H(x_m)=m{\int }_a^{b}F$ and $H(x_M)=M{\int }_a^{b}F$.
  • thus we know $H(x_m)\le {\int }_a^{b}FG\le H(x_M)$
• As ${\int }_a^{b}F(x)G(x)dx$ is between $H(x_m)$ and $H(x_M)$ and WLOG $x_m<x_M$, by the ZWS $\exists c$ s.t. $H(c)={\int }_a^{b}F(x)G(x)dx$, and $H(c)=G(c){\int }_a^{b}F(x)dx$.