Mixing
Is a function less than a decreasing function also decreasing?
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Two families of functions, $f_alpha(x)$ and $g_alpha(x)$, where $0leqalpha, xleq 1$.
It is known that $f_alpha(0)=g_alpha(0)=1$ and $f_alpha(1)=g_alpha(1)=0$ for any $alpha$.
Also, $f_0(x)=g_0(x)leq 1$ and $f_1(x)=g_1(x)=1$ for any $xin[0, 1]$.
Knowing that $g_alpha(x)$ decreases in $x$ for any $alpha$, can we say the same for $f_alpha$?
real-analysis calculus functions
edited Jan 13 at 15:06
daw
24.3k1645
asked Jan 12 at 20:00
HHWWHHWW
62
$endgroup$
add a comment |
$begingroup$
Two families of functions, $f_alpha(x)$ and $g_alpha(x)$, where $0leqalpha, xleq 1$.
It is known that $f_alpha(0)=g_alpha(0)=1$ and $f_alpha(1)=g_alpha(1)=0$ for any $alpha$.
Also, $f_0(x)=g_0(x)leq 1$ and $f_1(x)=g_1(x)=1$ for any $xin[0, 1]$.
Knowing that $g_alpha(x)$ decreases in $x$ for any $alpha$, can we say the same for $f_alpha$?
real-analysis calculus functions
edited Jan 13 at 15:06
daw
24.3k1645
asked Jan 12 at 20:00
HHWWHHWW
62
$endgroup$
2
$begingroup$
Try drawing this out; you should be able to come up with an example where $f$ decreases at a faster rate in one region, then increases for a bit, then decreases again, all while being below $g$.
$endgroup$
– jonem
Jan 12 at 20:03
add a comment |
$begingroup$
Two families of functions, $f_alpha(x)$ and $g_alpha(x)$, where $0leqalpha, xleq 1$.
It is known that $f_alpha(0)=g_alpha(0)=1$ and $f_alpha(1)=g_alpha(1)=0$ for any $alpha$.
Also, $f_0(x)=g_0(x)leq 1$ and $f_1(x)=g_1(x)=1$ for any $xin[0, 1]$.
Knowing that $g_alpha(x)$ decreases in $x$ for any $alpha$, can we say the same for $f_alpha$?
real-analysis calculus functions
edited Jan 13 at 15:06
daw
24.3k1645
asked Jan 12 at 20:00
HHWWHHWW
62
$endgroup$
Two families of functions, $f_alpha(x)$ and $g_alpha(x)$, where $0leqalpha, xleq 1$.
It is known that $f_alpha(0)=g_alpha(0)=1$ and $f_alpha(1)=g_alpha(1)=0$ for any $alpha$.
Also, $f_0(x)=g_0(x)leq 1$ and $f_1(x)=g_1(x)=1$ for any $xin[0, 1]$.
Knowing that $g_alpha(x)$ decreases in $x$ for any $alpha$, can we say the same for $f_alpha$?
real-analysis calculus functions
real-analysis calculus functions
edited Jan 13 at 15:06
daw
24.3k1645
asked Jan 12 at 20:00
HHWWHHWW
62
edited Jan 13 at 15:06
daw
24.3k1645
asked Jan 12 at 20:00
HHWWHHWW
62
edited Jan 13 at 15:06
daw
24.3k1645
edited Jan 13 at 15:06
daw
24.3k1645
edited Jan 13 at 15:06
daw
24.3k1645
24.3k1645
asked Jan 12 at 20:00
HHWWHHWW
62
asked Jan 12 at 20:00
HHWWHHWW
62
asked Jan 12 at 20:00
HHWWHHWW
62
62
2
$begingroup$
Try drawing this out; you should be able to come up with an example where $f$ decreases at a faster rate in one region, then increases for a bit, then decreases again, all while being below $g$.
$endgroup$
– jonem
Jan 12 at 20:03
add a comment |
2
$begingroup$
Try drawing this out; you should be able to come up with an example where $f$ decreases at a faster rate in one region, then increases for a bit, then decreases again, all while being below $g$.
$endgroup$
– jonem
Jan 12 at 20:03
2
2
$begingroup$
Try drawing this out; you should be able to come up with an example where $f$ decreases at a faster rate in one region, then increases for a bit, then decreases again, all while being below $g$.
$endgroup$
– jonem
Jan 12 at 20:03
$begingroup$
Try drawing this out; you should be able to come up with an example where $f$ decreases at a faster rate in one region, then increases for a bit, then decreases again, all while being below $g$.
$endgroup$
– jonem
Jan 12 at 20:03
add a comment |
1 Answer
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$begingroup$
Here we have $fle g$ and $g$ is decreasing but, as you can see from the picture, nothing can be deduced regarding if $f$ is increasing or decreasing.
answered Jan 12 at 20:13
BigbearZzzBigbearZzz
8,69121652
$endgroup$
add a comment |
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$begingroup$
Here we have $fle g$ and $g$ is decreasing but, as you can see from the picture, nothing can be deduced regarding if $f$ is increasing or decreasing.
answered Jan 12 at 20:13
BigbearZzzBigbearZzz
8,69121652
$endgroup$
add a comment |
$begingroup$
Here we have $fle g$ and $g$ is decreasing but, as you can see from the picture, nothing can be deduced regarding if $f$ is increasing or decreasing.
answered Jan 12 at 20:13
BigbearZzzBigbearZzz
8,69121652
$endgroup$
add a comment |
$begingroup$
Here we have $fle g$ and $g$ is decreasing but, as you can see from the picture, nothing can be deduced regarding if $f$ is increasing or decreasing.
answered Jan 12 at 20:13
BigbearZzzBigbearZzz
8,69121652
$endgroup$
Here we have $fle g$ and $g$ is decreasing but, as you can see from the picture, nothing can be deduced regarding if $f$ is increasing or decreasing.
answered Jan 12 at 20:13
BigbearZzzBigbearZzz
8,69121652
answered Jan 12 at 20:13
BigbearZzzBigbearZzz
8,69121652
answered Jan 12 at 20:13
BigbearZzzBigbearZzz
8,69121652
answered Jan 12 at 20:13
BigbearZzzBigbearZzz
8,69121652
8,69121652
add a comment |
add a comment |
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$begingroup$
Try drawing this out; you should be able to come up with an example where $f$ decreases at a faster rate in one region, then increases for a bit, then decreases again, all while being below $g$.
$endgroup$
– jonem
Jan 12 at 20:03