Prove that $\exists a<b$ s.t. $f(a)=f(b)=0$ when $\int_0^1f(x)dx=\int_0^1xf(x)dx=0$

The exact question is:

$f:[0,1]\rightarrow\mathbb{R}$ is a continuous function such that $\int_0^1f(x)=\int_0^1xf(x)=0$. Prove that $\exists a,b \in [0,1], a<b$ such that $f(a)=f(b)=0$.

By using mean value theorem of integration, we get some $a,b$ s.t $f(a)=f(b)=0$, but how can we show that $a\neq b$.

We are given that $\displaystyle \int^1_0 (ax+b)f(x) dx =0 .$ Suppose $f$ is not identically zero for otherwise the result is trivial. The condition $\displaystyle \int^1_0 f(x) dx = 0$ implies there is at least one sign-changing root, say at $m$ (that is, the function has different signs after passing through $m$, or more formally, $f(m+\epsilon)f(m-\epsilon)< 0$ for all sufficiently small $\epsilon>0.$) Suppose this is the only sign-changing root. Then $(x-m)f(x)$ does not change signs and is not identically $0$ either, so $\int^1_0 (x-m) f(x) \neq 0$, contradicting the first statement. Thus there are at least $2$ distinct sign-changing roots.
This idea can be employed in proving the generalized result: If $f\in C( [0,1], \mathbb{R} )$ is such that $\displaystyle \int^1_0 x^k f(x) dx = 0$ for all $k= 0, 1,2,\cdots, n-1$ then $f$ has at least $n$ distinct sign-changing roots in $[0,1].$
• Pls clarify.when $x<m,(x-m)<0$ and viceversa for $x>m$.So we get same sign for $(x-m)f(x)$. How can we argue for $xf(x)$ or for $(x-m)^2f(x)$ – bharath Feb 22 '12 at 18:02
• Since $(x-m) f(x)$ has the same sign throughout $[0,1]$ and is not identically zero, we must have $\left| \int^1_0 (x-m)f(x) dx \right| >0.$ But this contradicts $\int^1_0 (x-m) f(x) dx = \int^1_0 x f(x) dx - m \int^1_0 f(x) dx = 0-m\cdot 0 = 0.$ – Ragib Zaman Feb 23 '12 at 0:09
The text passage contains a mathematical question and a detailed solution to that question, including a proof and a discussion of a generalized result. There is also a clarifying question and an answer to that question.
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    1. If it contains a mathematical question (including word problems or calculations)
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First, describe the content structure you found in 1-2 sentences.

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TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Prove that $\exists a<b$ s.t. $f(a)=f(b)=0$ when $\int_0^1f(x)dx=\int_0^1xf(x)dx=0$

The exact question is:

$f:[0,1]\rightarrow\mathbb{R}$ is a continuous function such that $\int_0^1f(x)=\int_0^1xf(x)=0$. Prove that $\exists a,b \in [0,1], a<b$ such that $f(a)=f(b)=0$.

By using mean value theorem of integration, we get some $a,b$ s.t $f(a)=f(b)=0$, but how can we show that $a\neq b$.

We are given that $\displaystyle \int^1_0 (ax+b)f(x) dx =0 .$ Suppose $f$ is not identically zero for otherwise the result is trivial. The condition $\displaystyle \int^1_0 f(x) dx = 0$ implies there is at least one sign-changing root, say at $m$ (that is, the function has different signs after passing through $m$, or more formally, $f(m+\epsilon)f(m-\epsilon)< 0$ for all sufficiently small $\epsilon>0.$) Suppose this is the only sign-changing root. Then $(x-m)f(x)$ does not change signs and is not identically $0$ either, so $\int^1_0 (x-m) f(x) \neq 0$, contradicting the first statement. Thus there are at least $2$ distinct sign-changing roots.
This idea can be employed in proving the generalized result: If $f\in C( [0,1], \mathbb{R} )$ is such that $\displaystyle \int^1_0 x^k f(x) dx = 0$ for all $k= 0, 1,2,\cdots, n-1$ then $f$ has at least $n$ distinct sign-changing roots in $[0,1].$
• Pls clarify.when $x<m,(x-m)<0$ and viceversa for $x>m$.So we get same sign for $(x-m)f(x)$. How can we argue for $xf(x)$ or for $(x-m)^2f(x)$ – bharath Feb 22 '12 at 18:02
• Since $(x-m) f(x)$ has the same sign throughout $[0,1]$ and is not identically zero, we must have $\left| \int^1_0 (x-m)f(x) dx \right| >0.$ But this contradicts $\int^1_0 (x-m) f(x) dx = \int^1_0 x f(x) dx - m \int^1_0 f(x) dx = 0-m\cdot 0 = 0.$ – Ragib Zaman Feb 23 '12 at 0:09
user
The text passage contains a mathematical question and a detailed solution to that question, including a proof and a discussion of a generalized result. There is also a clarifying question and an answer to that question.

TYPE: SINGLE_QUESTION_WITH_ANSWER