Riemann Integration Problem

Question

Let $f$ be a continuous function on $[0,1]$ satisfying $$\int_0^1f(x)\,dx = 0$$

and $$\int_0^1xf(x)\,dx = 0.$$ Show that there exists $a$,$b$ in $[0,1]$ with $a < b$, such that $f(a) = 0 =f(b)$. Existence of one point is clear to me but I cannot prove the existence of the other one.

Thanks for any help.

You accept an answer by clicking on the check mark $\sqrt{}$ below the vertical arrows used for voting. — Michael Hardy, May 23 '12 at 16:39

score 7 · Accepted Answer · answered May 23 '12 at 14:31

7

If there was only one root $a$, unless $f$ is identically zero $f$ would have to change signs at $a$ in order for $\displaystyle\int^1_0 f(x) dx=0$ to hold. Hence $(x-a)f(x)$ doesn't change sign and is not identically zero, so $\displaystyle\int^1_0 (x-a) f(x) \neq 0,$ contradicting the hypothesis.

answered May 23 '12 at 14:31

Ragib Zaman

35,127

1

Thank you very much Ragib for such a nice proof – Ester May 24 '12 at 16:43

Riemann Integration Problem

1 Answers1