Show the relationship between the supremum and infimum of f^2 and |f|
Suppose f: [a,b] $to$ $mathbb{R}$ and B satisfy |f(x)| $le$ B for every x $epsilon$ [a,b].
Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then
M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $le$
2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))
for every 1 $le$ i $le$ n.
We are given a hint, namely that
|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.
And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.
real-analysis riemann-integration riemann-sum
asked Nov 22 '18 at 2:49
kendal
337
add a comment |
Suppose f: [a,b] $to$ $mathbb{R}$ and B satisfy |f(x)| $le$ B for every x $epsilon$ [a,b].
Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then
M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $le$
2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))
for every 1 $le$ i $le$ n.
We are given a hint, namely that
|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.
And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.
real-analysis riemann-integration riemann-sum
asked Nov 22 '18 at 2:49
kendal
337
What are the functions $M$ and $m$?
– Sean Roberson
Nov 22 '18 at 2:51
Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
– kendal
Nov 22 '18 at 2:55
4
The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
– Paramanand Singh
Nov 22 '18 at 4:18
add a comment |
Suppose f: [a,b] $to$ $mathbb{R}$ and B satisfy |f(x)| $le$ B for every x $epsilon$ [a,b].
Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then
M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $le$
2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))
for every 1 $le$ i $le$ n.
We are given a hint, namely that
|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.
And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.
real-analysis riemann-integration riemann-sum
asked Nov 22 '18 at 2:49
kendal
337
Suppose f: [a,b] $to$ $mathbb{R}$ and B satisfy |f(x)| $le$ B for every x $epsilon$ [a,b].
Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then
M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $le$
2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))
for every 1 $le$ i $le$ n.
We are given a hint, namely that
|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.
And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.
real-analysis riemann-integration riemann-sum
real-analysis riemann-integration riemann-sum
asked Nov 22 '18 at 2:49
kendal
337
asked Nov 22 '18 at 2:49
kendal
337
edited Nov 22 '18 at 3:05
asked Nov 22 '18 at 2:49
kendal
337
asked Nov 22 '18 at 2:49
kendal
337
asked Nov 22 '18 at 2:49
kendal
337
337
What are the functions $M$ and $m$?
– Sean Roberson
Nov 22 '18 at 2:51
Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
– kendal
Nov 22 '18 at 2:55
4
The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
– Paramanand Singh
Nov 22 '18 at 4:18
add a comment |
What are the functions $M$ and $m$?
– Sean Roberson
Nov 22 '18 at 2:51
Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
– kendal
Nov 22 '18 at 2:55
4
The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
– Paramanand Singh
Nov 22 '18 at 4:18
What are the functions $M$ and $m$?
– Sean Roberson
Nov 22 '18 at 2:51
What are the functions $M$ and $m$?
– Sean Roberson
Nov 22 '18 at 2:51
Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
– kendal
Nov 22 '18 at 2:55
Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
– kendal
Nov 22 '18 at 2:55
4
4
The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
– Paramanand Singh
Nov 22 '18 at 4:18
The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
– Paramanand Singh
Nov 22 '18 at 4:18
add a comment |
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What are the functions $M$ and $m$?
– Sean Roberson
Nov 22 '18 at 2:51
Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
– kendal
Nov 22 '18 at 2:55
4
The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
– Paramanand Singh
Nov 22 '18 at 4:18