Mixing
Let $S = { x in R : x^{6} - x^{5} ≤ 100} $ and $T = { x^{2}-2x : x in (0, infty)} $..
$begingroup$
Let $S = { x in R : x^{6} - x^{5} ≤ 100} $ and $T = { x^{2}-2x : x in (0, infty)} $ Then the set $S$ intersection $T$ is closed and bounded. (True/false)
The range of set $T$ is $[-1, infty]$ which is closed.
I need to find set $S$ but I don't know how to do it. I can use the sign scheme of polynomial $x^{6} -x^{5} - 100$ to see where the value of polynomial is negative. But finding out its roots is not an easy task
How to solve this problem?
calculus
asked Jan 22 at 7:38
MathsaddictMathsaddict
3669
$endgroup$
add a comment |
$begingroup$
Let $S = { x in R : x^{6} - x^{5} ≤ 100} $ and $T = { x^{2}-2x : x in (0, infty)} $ Then the set $S$ intersection $T$ is closed and bounded. (True/false)
The range of set $T$ is $[-1, infty]$ which is closed.
I need to find set $S$ but I don't know how to do it. I can use the sign scheme of polynomial $x^{6} -x^{5} - 100$ to see where the value of polynomial is negative. But finding out its roots is not an easy task
How to solve this problem?
calculus
asked Jan 22 at 7:38
MathsaddictMathsaddict
3669
$endgroup$
add a comment |
$begingroup$
Let $S = { x in R : x^{6} - x^{5} ≤ 100} $ and $T = { x^{2}-2x : x in (0, infty)} $ Then the set $S$ intersection $T$ is closed and bounded. (True/false)
The range of set $T$ is $[-1, infty]$ which is closed.
I need to find set $S$ but I don't know how to do it. I can use the sign scheme of polynomial $x^{6} -x^{5} - 100$ to see where the value of polynomial is negative. But finding out its roots is not an easy task
How to solve this problem?
calculus
asked Jan 22 at 7:38
MathsaddictMathsaddict
3669
$endgroup$
Let $S = { x in R : x^{6} - x^{5} ≤ 100} $ and $T = { x^{2}-2x : x in (0, infty)} $ Then the set $S$ intersection $T$ is closed and bounded. (True/false)
The range of set $T$ is $[-1, infty]$ which is closed.
I need to find set $S$ but I don't know how to do it. I can use the sign scheme of polynomial $x^{6} -x^{5} - 100$ to see where the value of polynomial is negative. But finding out its roots is not an easy task
How to solve this problem?
calculus
calculus
asked Jan 22 at 7:38
MathsaddictMathsaddict
3669
asked Jan 22 at 7:38
MathsaddictMathsaddict
3669
edited Jan 22 at 7:42
Mathsaddict
asked Jan 22 at 7:38
MathsaddictMathsaddict
3669
asked Jan 22 at 7:38
MathsaddictMathsaddict
3669
asked Jan 22 at 7:38
MathsaddictMathsaddict
3669
3669
add a comment |
add a comment |
1 Answer
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$begingroup$
$S$ is certainly closed as $le$ is a "closed condition" (formally, pre-image o fthe closed set $(-infty,100]$ under the continuous map $xmapsto x^ 6-x^5$). Henec $Scap T$ is closed.
We already know that $Scap T$ is bounded from below and only need to check for an upper bound. Note that the function $xmapsto x^6-x^5$ goes to $+infty$ as $xtopminfty$ because the $x^6$ is dominating. Hence for $|x|$ large enough, $x^6-x^5$ exceeds any given limit, such as $100$. We conclude that $S$ is bounded.
answered Jan 22 at 7:42
Hagen von EitzenHagen von Eitzen
282k23272507
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
$S$ is certainly closed as $le$ is a "closed condition" (formally, pre-image o fthe closed set $(-infty,100]$ under the continuous map $xmapsto x^ 6-x^5$). Henec $Scap T$ is closed.
We already know that $Scap T$ is bounded from below and only need to check for an upper bound. Note that the function $xmapsto x^6-x^5$ goes to $+infty$ as $xtopminfty$ because the $x^6$ is dominating. Hence for $|x|$ large enough, $x^6-x^5$ exceeds any given limit, such as $100$. We conclude that $S$ is bounded.
answered Jan 22 at 7:42
Hagen von EitzenHagen von Eitzen
282k23272507
$endgroup$
add a comment |
$begingroup$
$S$ is certainly closed as $le$ is a "closed condition" (formally, pre-image o fthe closed set $(-infty,100]$ under the continuous map $xmapsto x^ 6-x^5$). Henec $Scap T$ is closed.
We already know that $Scap T$ is bounded from below and only need to check for an upper bound. Note that the function $xmapsto x^6-x^5$ goes to $+infty$ as $xtopminfty$ because the $x^6$ is dominating. Hence for $|x|$ large enough, $x^6-x^5$ exceeds any given limit, such as $100$. We conclude that $S$ is bounded.
answered Jan 22 at 7:42
Hagen von EitzenHagen von Eitzen
282k23272507
$endgroup$
add a comment |
$begingroup$
$S$ is certainly closed as $le$ is a "closed condition" (formally, pre-image o fthe closed set $(-infty,100]$ under the continuous map $xmapsto x^ 6-x^5$). Henec $Scap T$ is closed.
We already know that $Scap T$ is bounded from below and only need to check for an upper bound. Note that the function $xmapsto x^6-x^5$ goes to $+infty$ as $xtopminfty$ because the $x^6$ is dominating. Hence for $|x|$ large enough, $x^6-x^5$ exceeds any given limit, such as $100$. We conclude that $S$ is bounded.
answered Jan 22 at 7:42
Hagen von EitzenHagen von Eitzen
282k23272507
$endgroup$
$S$ is certainly closed as $le$ is a "closed condition" (formally, pre-image o fthe closed set $(-infty,100]$ under the continuous map $xmapsto x^ 6-x^5$). Henec $Scap T$ is closed.
We already know that $Scap T$ is bounded from below and only need to check for an upper bound. Note that the function $xmapsto x^6-x^5$ goes to $+infty$ as $xtopminfty$ because the $x^6$ is dominating. Hence for $|x|$ large enough, $x^6-x^5$ exceeds any given limit, such as $100$. We conclude that $S$ is bounded.
answered Jan 22 at 7:42
Hagen von EitzenHagen von Eitzen
282k23272507
answered Jan 22 at 7:42
Hagen von EitzenHagen von Eitzen
282k23272507
answered Jan 22 at 7:42
Hagen von EitzenHagen von Eitzen
282k23272507
answered Jan 22 at 7:42
Hagen von EitzenHagen von Eitzen
282k23272507
282k23272507
add a comment |
add a comment |
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