Let $f$ be a continuous function on an interval $[a, b]$. If for every function $g$ on the interval $[a, b]$ such $g(a)=g(b)=0$ and $\displaystyle \int_a^b f(x)g(x)\, {\rm d}x=0$ hold , then prove that $f$ is the zero function.
Solution
We consider the function $g(x)=(x-a)(b-x)f(x)$ and from the hypothesis holds:
$$\int_{a}^{b}(x-a)(b-x)f^2(x)\, {\rm d}x =0$$
Since $(x-a)(b-x)f^2(x) \geq 0, \; \forall x \in [a,b]$ (product of non negative terms) then $(x-a)(b-x)f^2(x)=0 ,\; \forall x \in [a,b]$. More specifically $f(x)=0 , \; x \in (a, b)$. From continuity $f$ is also zero at the limits of the interval and the result follows.
The exercise can also be found in mathematica.gr
Solution
We consider the function $g(x)=(x-a)(b-x)f(x)$ and from the hypothesis holds:
$$\int_{a}^{b}(x-a)(b-x)f^2(x)\, {\rm d}x =0$$
Since $(x-a)(b-x)f^2(x) \geq 0, \; \forall x \in [a,b]$ (product of non negative terms) then $(x-a)(b-x)f^2(x)=0 ,\; \forall x \in [a,b]$. More specifically $f(x)=0 , \; x \in (a, b)$. From continuity $f$ is also zero at the limits of the interval and the result follows.
The exercise can also be found in mathematica.gr
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