Proving existence of a global maximum
I am given a function $f$ that is continuous and periodic with period $p$. The domain of the function is entire real numbers. I have to prove that the function has a global maximum.
Attempt:
First, I proved that a continuous function that is bounded by two elements on its domain must have a maximum.
Question:
Is it correct to now just write that since $f$ is periodic, if I consider the function in an interval $[0,p]$, then the function will just be repeated outside the interval and since I have proved that there is a maximum in that interval, it will also be the global maximum? What I am confused about is: the question asks me to prove existence of a global maximum. But if the function is repeating with period $p$, there are many such maxima. But isn't global maximum unique? What am I understanding wrong?
real-analysis
edited Mar 21 at 19:43
Leyla Alkan
1,5411723
asked Mar 21 at 19:41
Ufomammut
258313
add a comment |
I am given a function $f$ that is continuous and periodic with period $p$. The domain of the function is entire real numbers. I have to prove that the function has a global maximum.
Attempt:
First, I proved that a continuous function that is bounded by two elements on its domain must have a maximum.
Question:
Is it correct to now just write that since $f$ is periodic, if I consider the function in an interval $[0,p]$, then the function will just be repeated outside the interval and since I have proved that there is a maximum in that interval, it will also be the global maximum? What I am confused about is: the question asks me to prove existence of a global maximum. But if the function is repeating with period $p$, there are many such maxima. But isn't global maximum unique? What am I understanding wrong?
real-analysis
edited Mar 21 at 19:43
Leyla Alkan
1,5411723
asked Mar 21 at 19:41
Ufomammut
258313
2
A global maxima is unique but can be reached by multiple points. Take $f(x)=sin(x)$, it is $2pi$-periodic and has a global maxima of $1$ but you reach this global maxima for an infinite amount of points of the form ${piover 2}+2kpi, kinmathbb{N}$
– Furrane
Mar 21 at 19:47
add a comment |
I am given a function $f$ that is continuous and periodic with period $p$. The domain of the function is entire real numbers. I have to prove that the function has a global maximum.
Attempt:
First, I proved that a continuous function that is bounded by two elements on its domain must have a maximum.
Question:
Is it correct to now just write that since $f$ is periodic, if I consider the function in an interval $[0,p]$, then the function will just be repeated outside the interval and since I have proved that there is a maximum in that interval, it will also be the global maximum? What I am confused about is: the question asks me to prove existence of a global maximum. But if the function is repeating with period $p$, there are many such maxima. But isn't global maximum unique? What am I understanding wrong?
real-analysis
edited Mar 21 at 19:43
Leyla Alkan
1,5411723
asked Mar 21 at 19:41
Ufomammut
258313
I am given a function $f$ that is continuous and periodic with period $p$. The domain of the function is entire real numbers. I have to prove that the function has a global maximum.
Attempt:
First, I proved that a continuous function that is bounded by two elements on its domain must have a maximum.
Question:
Is it correct to now just write that since $f$ is periodic, if I consider the function in an interval $[0,p]$, then the function will just be repeated outside the interval and since I have proved that there is a maximum in that interval, it will also be the global maximum? What I am confused about is: the question asks me to prove existence of a global maximum. But if the function is repeating with period $p$, there are many such maxima. But isn't global maximum unique? What am I understanding wrong?
real-analysis
real-analysis
edited Mar 21 at 19:43
Leyla Alkan
1,5411723
asked Mar 21 at 19:41
Ufomammut
258313
edited Mar 21 at 19:43
Leyla Alkan
1,5411723
asked Mar 21 at 19:41
Ufomammut
258313
edited Mar 21 at 19:43
Leyla Alkan
1,5411723
edited Mar 21 at 19:43
Leyla Alkan
1,5411723
edited Mar 21 at 19:43
Leyla Alkan
1,5411723
1,5411723
asked Mar 21 at 19:41
Ufomammut
258313
asked Mar 21 at 19:41
Ufomammut
258313
asked Mar 21 at 19:41
Ufomammut
258313
258313
2
A global maxima is unique but can be reached by multiple points. Take $f(x)=sin(x)$, it is $2pi$-periodic and has a global maxima of $1$ but you reach this global maxima for an infinite amount of points of the form ${piover 2}+2kpi, kinmathbb{N}$
– Furrane
Mar 21 at 19:47
add a comment |
2
A global maxima is unique but can be reached by multiple points. Take $f(x)=sin(x)$, it is $2pi$-periodic and has a global maxima of $1$ but you reach this global maxima for an infinite amount of points of the form ${piover 2}+2kpi, kinmathbb{N}$
– Furrane
Mar 21 at 19:47
2
2
A global maxima is unique but can be reached by multiple points. Take $f(x)=sin(x)$, it is $2pi$-periodic and has a global maxima of $1$ but you reach this global maxima for an infinite amount of points of the form ${piover 2}+2kpi, kinmathbb{N}$
– Furrane
Mar 21 at 19:47
A global maxima is unique but can be reached by multiple points. Take $f(x)=sin(x)$, it is $2pi$-periodic and has a global maxima of $1$ but you reach this global maxima for an infinite amount of points of the form ${piover 2}+2kpi, kinmathbb{N}$
– Furrane
Mar 21 at 19:47
add a comment |
1 Answer
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From the comments above by @Furrane.
The global maximum is unique but it can be attained by multiple points. Take $f(x)=sin(x)$. It is $2pi$-periodic and has a global maximum equal to $1$. But, this maximum is attained by an infinite number of points, namely every point of the form $frac{pi}{2} + 2 k pi$, where $k in mathbb{Z}$.
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From the comments above by @Furrane.
The global maximum is unique but it can be attained by multiple points. Take $f(x)=sin(x)$. It is $2pi$-periodic and has a global maximum equal to $1$. But, this maximum is attained by an infinite number of points, namely every point of the form $frac{pi}{2} + 2 k pi$, where $k in mathbb{Z}$.
add a comment |
From the comments above by @Furrane.
The global maximum is unique but it can be attained by multiple points. Take $f(x)=sin(x)$. It is $2pi$-periodic and has a global maximum equal to $1$. But, this maximum is attained by an infinite number of points, namely every point of the form $frac{pi}{2} + 2 k pi$, where $k in mathbb{Z}$.
add a comment |
From the comments above by @Furrane.
The global maximum is unique but it can be attained by multiple points. Take $f(x)=sin(x)$. It is $2pi$-periodic and has a global maximum equal to $1$. But, this maximum is attained by an infinite number of points, namely every point of the form $frac{pi}{2} + 2 k pi$, where $k in mathbb{Z}$.
From the comments above by @Furrane.
The global maximum is unique but it can be attained by multiple points. Take $f(x)=sin(x)$. It is $2pi$-periodic and has a global maximum equal to $1$. But, this maximum is attained by an infinite number of points, namely every point of the form $frac{pi}{2} + 2 k pi$, where $k in mathbb{Z}$.
answered Nov 29 at 17:45
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Brahadeesh
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A global maxima is unique but can be reached by multiple points. Take $f(x)=sin(x)$, it is $2pi$-periodic and has a global maxima of $1$ but you reach this global maxima for an infinite amount of points of the form ${piover 2}+2kpi, kinmathbb{N}$
– Furrane
Mar 21 at 19:47