$f: [-1, 1] to (0, 1]$ is continuous. Prove that $f(x) = x^4$ has at least 2 roots in $[-1, 1]$ [on hold]
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Let $f: [-1, 1] to (0, 1]$ be a continuous function. Prove that the equation $f(x) = x^4$ has at least $2$ roots in $[-1, 1]$ using intermediate value theorem.
It is intuitively obvious, but I don't know how to prove it formally.
continuity roots
edited Jan 7 at 10:47
Robert Z
94.5k1062133
asked Jan 7 at 7:36
Quo Si ThanQuo Si Than
1437
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put on hold as off-topic by RRL, Holo, Alexander Gruber♦ yesterday
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$begingroup$
Let $f: [-1, 1] to (0, 1]$ be a continuous function. Prove that the equation $f(x) = x^4$ has at least $2$ roots in $[-1, 1]$ using intermediate value theorem.
It is intuitively obvious, but I don't know how to prove it formally.
continuity roots
edited Jan 7 at 10:47
Robert Z
94.5k1062133
asked Jan 7 at 7:36
Quo Si ThanQuo Si Than
1437
$endgroup$
put on hold as off-topic by RRL, Holo, Alexander Gruber♦ yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Holo, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Let $f: [-1, 1] to (0, 1]$ be a continuous function. Prove that the equation $f(x) = x^4$ has at least $2$ roots in $[-1, 1]$ using intermediate value theorem.
It is intuitively obvious, but I don't know how to prove it formally.
continuity roots
edited Jan 7 at 10:47
Robert Z
94.5k1062133
asked Jan 7 at 7:36
Quo Si ThanQuo Si Than
1437
$endgroup$
Let $f: [-1, 1] to (0, 1]$ be a continuous function. Prove that the equation $f(x) = x^4$ has at least $2$ roots in $[-1, 1]$ using intermediate value theorem.
It is intuitively obvious, but I don't know how to prove it formally.
continuity roots
continuity roots
edited Jan 7 at 10:47
Robert Z
94.5k1062133
asked Jan 7 at 7:36
Quo Si ThanQuo Si Than
1437
edited Jan 7 at 10:47
Robert Z
94.5k1062133
asked Jan 7 at 7:36
Quo Si ThanQuo Si Than
1437
edited Jan 7 at 10:47
Robert Z
94.5k1062133
edited Jan 7 at 10:47
Robert Z
94.5k1062133
edited Jan 7 at 10:47
Robert Z
94.5k1062133
94.5k1062133
asked Jan 7 at 7:36
Quo Si ThanQuo Si Than
1437
asked Jan 7 at 7:36
Quo Si ThanQuo Si Than
1437
asked Jan 7 at 7:36
Quo Si ThanQuo Si Than
1437
1437
put on hold as off-topic by RRL, Holo, Alexander Gruber♦ yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Holo, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as off-topic by RRL, Holo, Alexander Gruber♦ yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Holo, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
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2 Answers
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Hint. Let $F(x)=f(x)-x^4$ and note that $F$ is a continuous function such that $$F(-1)=f(-1)-1leq 0,quad F(0)=f(0)>0, text{ and } F(1)=f(1)-1leq 0.$$
Then use the intermediate value theorem.
answered Jan 7 at 7:46
Robert ZRobert Z
94.5k1062133
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add a comment |
$begingroup$
Let $g(x):=f(x)-x^4$.
Then $g(-1)=f(-1)-1 le 0$ and $g(0)=f(0)>0$. Thus there is $x_1 in [-1,0)$ such that $g(x_1)=0$.
Similar arguments show that there ist $x_2 in (0,1]$ with $g(x_2)=0$.
answered Jan 7 at 7:47
FredFred
44.4k1845
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add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint. Let $F(x)=f(x)-x^4$ and note that $F$ is a continuous function such that $$F(-1)=f(-1)-1leq 0,quad F(0)=f(0)>0, text{ and } F(1)=f(1)-1leq 0.$$
Then use the intermediate value theorem.
answered Jan 7 at 7:46
Robert ZRobert Z
94.5k1062133
$endgroup$
add a comment |
$begingroup$
Hint. Let $F(x)=f(x)-x^4$ and note that $F$ is a continuous function such that $$F(-1)=f(-1)-1leq 0,quad F(0)=f(0)>0, text{ and } F(1)=f(1)-1leq 0.$$
Then use the intermediate value theorem.
answered Jan 7 at 7:46
Robert ZRobert Z
94.5k1062133
$endgroup$
add a comment |
$begingroup$
Hint. Let $F(x)=f(x)-x^4$ and note that $F$ is a continuous function such that $$F(-1)=f(-1)-1leq 0,quad F(0)=f(0)>0, text{ and } F(1)=f(1)-1leq 0.$$
Then use the intermediate value theorem.
answered Jan 7 at 7:46
Robert ZRobert Z
94.5k1062133
$endgroup$
Hint. Let $F(x)=f(x)-x^4$ and note that $F$ is a continuous function such that $$F(-1)=f(-1)-1leq 0,quad F(0)=f(0)>0, text{ and } F(1)=f(1)-1leq 0.$$
Then use the intermediate value theorem.
answered Jan 7 at 7:46
Robert ZRobert Z
94.5k1062133
edited Jan 7 at 7:54
answered Jan 7 at 7:46
Robert ZRobert Z
94.5k1062133
answered Jan 7 at 7:46
Robert ZRobert Z
94.5k1062133
answered Jan 7 at 7:46
Robert ZRobert Z
94.5k1062133
94.5k1062133
add a comment |
add a comment |
$begingroup$
Let $g(x):=f(x)-x^4$.
Then $g(-1)=f(-1)-1 le 0$ and $g(0)=f(0)>0$. Thus there is $x_1 in [-1,0)$ such that $g(x_1)=0$.
Similar arguments show that there ist $x_2 in (0,1]$ with $g(x_2)=0$.
answered Jan 7 at 7:47
FredFred
44.4k1845
$endgroup$
add a comment |
$begingroup$
Let $g(x):=f(x)-x^4$.
Then $g(-1)=f(-1)-1 le 0$ and $g(0)=f(0)>0$. Thus there is $x_1 in [-1,0)$ such that $g(x_1)=0$.
Similar arguments show that there ist $x_2 in (0,1]$ with $g(x_2)=0$.
answered Jan 7 at 7:47
FredFred
44.4k1845
$endgroup$
add a comment |
$begingroup$
Let $g(x):=f(x)-x^4$.
Then $g(-1)=f(-1)-1 le 0$ and $g(0)=f(0)>0$. Thus there is $x_1 in [-1,0)$ such that $g(x_1)=0$.
Similar arguments show that there ist $x_2 in (0,1]$ with $g(x_2)=0$.
answered Jan 7 at 7:47
FredFred
44.4k1845
$endgroup$
Let $g(x):=f(x)-x^4$.
Then $g(-1)=f(-1)-1 le 0$ and $g(0)=f(0)>0$. Thus there is $x_1 in [-1,0)$ such that $g(x_1)=0$.
Similar arguments show that there ist $x_2 in (0,1]$ with $g(x_2)=0$.
answered Jan 7 at 7:47
FredFred
44.4k1845
answered Jan 7 at 7:47
FredFred
44.4k1845
answered Jan 7 at 7:47
FredFred
44.4k1845
answered Jan 7 at 7:47
FredFred
44.4k1845
44.4k1845
add a comment |
add a comment |
