Does $int_0^1 f(x)=int_0^1 xf(x)$ imply $int_0^x f(t)$ has a root
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My question is whether or not the following is true:
If $f:[0,1]to mathbb{R}$ is a continous function such that
$$int_0^1 f(x)dx=int_0^1 xf(x)dx$$
then there exist $cin(0,1)$ such that
$$int_0^c f(x)dx=0$$
It is quite clear that in the interval $(0,1)$ there must be some points $a$ such that $f(1)f(a)<0$ but I can't say from this if the statement above is true or not.
real-analysis definite-integrals
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add a comment |
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My question is whether or not the following is true:
If $f:[0,1]to mathbb{R}$ is a continous function such that
$$int_0^1 f(x)dx=int_0^1 xf(x)dx$$
then there exist $cin(0,1)$ such that
$$int_0^c f(x)dx=0$$
It is quite clear that in the interval $(0,1)$ there must be some points $a$ such that $f(1)f(a)<0$ but I can't say from this if the statement above is true or not.
real-analysis definite-integrals
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Do you have an example of a non-zero function that satisfies the first equality?
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– Yanko
Jan 5 at 12:27
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@Yanko $operatorname{sinc}(2pi (1-x) )$ works
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– Calvin Khor
Jan 5 at 12:35
1
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@Yanko Surley there are such functions. There exist $a,b$ such that $ax+b$ satisfies that equation.
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– Kavi Rama Murthy
Jan 5 at 12:35
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@KaviRamaMurthy Right, for $a=6,b=-2$. Thanks.
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– Yanko
Jan 5 at 12:37
add a comment |
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My question is whether or not the following is true:
If $f:[0,1]to mathbb{R}$ is a continous function such that
$$int_0^1 f(x)dx=int_0^1 xf(x)dx$$
then there exist $cin(0,1)$ such that
$$int_0^c f(x)dx=0$$
It is quite clear that in the interval $(0,1)$ there must be some points $a$ such that $f(1)f(a)<0$ but I can't say from this if the statement above is true or not.
real-analysis definite-integrals
$endgroup$
My question is whether or not the following is true:
If $f:[0,1]to mathbb{R}$ is a continous function such that
$$int_0^1 f(x)dx=int_0^1 xf(x)dx$$
then there exist $cin(0,1)$ such that
$$int_0^c f(x)dx=0$$
It is quite clear that in the interval $(0,1)$ there must be some points $a$ such that $f(1)f(a)<0$ but I can't say from this if the statement above is true or not.
real-analysis definite-integrals
real-analysis definite-integrals
asked Jan 5 at 12:18
Andrew VAndrew V
31411
31411
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Do you have an example of a non-zero function that satisfies the first equality?
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– Yanko
Jan 5 at 12:27
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@Yanko $operatorname{sinc}(2pi (1-x) )$ works
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– Calvin Khor
Jan 5 at 12:35
1
$begingroup$
@Yanko Surley there are such functions. There exist $a,b$ such that $ax+b$ satisfies that equation.
$endgroup$
– Kavi Rama Murthy
Jan 5 at 12:35
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@KaviRamaMurthy Right, for $a=6,b=-2$. Thanks.
$endgroup$
– Yanko
Jan 5 at 12:37
add a comment |
$begingroup$
Do you have an example of a non-zero function that satisfies the first equality?
$endgroup$
– Yanko
Jan 5 at 12:27
$begingroup$
@Yanko $operatorname{sinc}(2pi (1-x) )$ works
$endgroup$
– Calvin Khor
Jan 5 at 12:35
1
$begingroup$
@Yanko Surley there are such functions. There exist $a,b$ such that $ax+b$ satisfies that equation.
$endgroup$
– Kavi Rama Murthy
Jan 5 at 12:35
$begingroup$
@KaviRamaMurthy Right, for $a=6,b=-2$. Thanks.
$endgroup$
– Yanko
Jan 5 at 12:37
$begingroup$
Do you have an example of a non-zero function that satisfies the first equality?
$endgroup$
– Yanko
Jan 5 at 12:27
$begingroup$
Do you have an example of a non-zero function that satisfies the first equality?
$endgroup$
– Yanko
Jan 5 at 12:27
$begingroup$
@Yanko $operatorname{sinc}(2pi (1-x) )$ works
$endgroup$
– Calvin Khor
Jan 5 at 12:35
$begingroup$
@Yanko $operatorname{sinc}(2pi (1-x) )$ works
$endgroup$
– Calvin Khor
Jan 5 at 12:35
1
1
$begingroup$
@Yanko Surley there are such functions. There exist $a,b$ such that $ax+b$ satisfies that equation.
$endgroup$
– Kavi Rama Murthy
Jan 5 at 12:35
$begingroup$
@Yanko Surley there are such functions. There exist $a,b$ such that $ax+b$ satisfies that equation.
$endgroup$
– Kavi Rama Murthy
Jan 5 at 12:35
$begingroup$
@KaviRamaMurthy Right, for $a=6,b=-2$. Thanks.
$endgroup$
– Yanko
Jan 5 at 12:37
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@KaviRamaMurthy Right, for $a=6,b=-2$. Thanks.
$endgroup$
– Yanko
Jan 5 at 12:37
add a comment |
2 Answers
2
active
oldest
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Let $F(x)=int_0^{x} f(t), dt$. Then $F(1)=int_0^{1} f(t), dt=int_0^{1} tf(t), dt=tF(t)|_0^{1}-int_0^{1}F(t), dt$. This gives $int_0^{1}F(t), dt=0$. If $F$ has no zeros then it is strictly positive or strictly negative throughout and its integral cannot be $0$. Hence $F(c)=0$ for some $c$.
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add a comment |
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Hint: Note that
$$
int_0^1 F(x)dx =int_0^1left(int_0^x f(t)dtright)dx=int_0^1left(int_t^1dxright)f(t)dt= int_0^1 (1-t)f(t)dt
$$ where $F(x) = int_0^x f(t)dt$.
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Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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Let $F(x)=int_0^{x} f(t), dt$. Then $F(1)=int_0^{1} f(t), dt=int_0^{1} tf(t), dt=tF(t)|_0^{1}-int_0^{1}F(t), dt$. This gives $int_0^{1}F(t), dt=0$. If $F$ has no zeros then it is strictly positive or strictly negative throughout and its integral cannot be $0$. Hence $F(c)=0$ for some $c$.
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add a comment |
$begingroup$
Let $F(x)=int_0^{x} f(t), dt$. Then $F(1)=int_0^{1} f(t), dt=int_0^{1} tf(t), dt=tF(t)|_0^{1}-int_0^{1}F(t), dt$. This gives $int_0^{1}F(t), dt=0$. If $F$ has no zeros then it is strictly positive or strictly negative throughout and its integral cannot be $0$. Hence $F(c)=0$ for some $c$.
$endgroup$
add a comment |
$begingroup$
Let $F(x)=int_0^{x} f(t), dt$. Then $F(1)=int_0^{1} f(t), dt=int_0^{1} tf(t), dt=tF(t)|_0^{1}-int_0^{1}F(t), dt$. This gives $int_0^{1}F(t), dt=0$. If $F$ has no zeros then it is strictly positive or strictly negative throughout and its integral cannot be $0$. Hence $F(c)=0$ for some $c$.
$endgroup$
Let $F(x)=int_0^{x} f(t), dt$. Then $F(1)=int_0^{1} f(t), dt=int_0^{1} tf(t), dt=tF(t)|_0^{1}-int_0^{1}F(t), dt$. This gives $int_0^{1}F(t), dt=0$. If $F$ has no zeros then it is strictly positive or strictly negative throughout and its integral cannot be $0$. Hence $F(c)=0$ for some $c$.
answered Jan 5 at 12:30
Kavi Rama MurthyKavi Rama Murthy
65.1k42766
65.1k42766
add a comment |
add a comment |
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Hint: Note that
$$
int_0^1 F(x)dx =int_0^1left(int_0^x f(t)dtright)dx=int_0^1left(int_t^1dxright)f(t)dt= int_0^1 (1-t)f(t)dt
$$ where $F(x) = int_0^x f(t)dt$.
$endgroup$
add a comment |
$begingroup$
Hint: Note that
$$
int_0^1 F(x)dx =int_0^1left(int_0^x f(t)dtright)dx=int_0^1left(int_t^1dxright)f(t)dt= int_0^1 (1-t)f(t)dt
$$ where $F(x) = int_0^x f(t)dt$.
$endgroup$
add a comment |
$begingroup$
Hint: Note that
$$
int_0^1 F(x)dx =int_0^1left(int_0^x f(t)dtright)dx=int_0^1left(int_t^1dxright)f(t)dt= int_0^1 (1-t)f(t)dt
$$ where $F(x) = int_0^x f(t)dt$.
$endgroup$
Hint: Note that
$$
int_0^1 F(x)dx =int_0^1left(int_0^x f(t)dtright)dx=int_0^1left(int_t^1dxright)f(t)dt= int_0^1 (1-t)f(t)dt
$$ where $F(x) = int_0^x f(t)dt$.
answered Jan 5 at 12:30
SongSong
16.8k11042
16.8k11042
add a comment |
add a comment |
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$begingroup$
Do you have an example of a non-zero function that satisfies the first equality?
$endgroup$
– Yanko
Jan 5 at 12:27
$begingroup$
@Yanko $operatorname{sinc}(2pi (1-x) )$ works
$endgroup$
– Calvin Khor
Jan 5 at 12:35
1
$begingroup$
@Yanko Surley there are such functions. There exist $a,b$ such that $ax+b$ satisfies that equation.
$endgroup$
– Kavi Rama Murthy
Jan 5 at 12:35
$begingroup$
@KaviRamaMurthy Right, for $a=6,b=-2$. Thanks.
$endgroup$
– Yanko
Jan 5 at 12:37