$f:[0,1]tomathbb{R}$ continuous, differentiable in $(0,1)$ such that $f(0)=0$ and $f(1)=1$ then...
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I don't have any idea of how to proceed. The complete problem is:
If $f:[0,1]tomathbb{R}$ is a continuous function in $[0,1]$ and differentiable in (0,1) such that $f(0)=0$ and $f(1)=1$. Prove that for all $kinmathbb{N}$ exist different $x_1,...,x_kin (0,1)$ such that $$sum_{i=1}^{k}frac{1}{f'(x_i)}=k$$
I'll be grateful for your help.
calculus derivatives continuity
edited Jan 7 at 1:11
Clayton
18.9k33085
asked Jan 7 at 0:57
Raul_MFerRaul_MFer
453
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put on hold as off-topic by user21820, José Carlos Santos, amWhy, Did, RRL Jan 7 at 17:34
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." – user21820, José Carlos Santos, amWhy, Did, RRL
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I don't have any idea of how to proceed. The complete problem is:
If $f:[0,1]tomathbb{R}$ is a continuous function in $[0,1]$ and differentiable in (0,1) such that $f(0)=0$ and $f(1)=1$. Prove that for all $kinmathbb{N}$ exist different $x_1,...,x_kin (0,1)$ such that $$sum_{i=1}^{k}frac{1}{f'(x_i)}=k$$
I'll be grateful for your help.
calculus derivatives continuity
edited Jan 7 at 1:11
Clayton
18.9k33085
asked Jan 7 at 0:57
Raul_MFerRaul_MFer
453
$endgroup$
put on hold as off-topic by user21820, José Carlos Santos, amWhy, Did, RRL Jan 7 at 17:34
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." – user21820, José Carlos Santos, amWhy, Did, RRL
If this question can be reworded to fit the rules in the help center, please edit the question.
2
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math.stackexchange.com/questions/366379/…
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– yuanming luo
Jan 7 at 1:30
add a comment |
$begingroup$
I don't have any idea of how to proceed. The complete problem is:
If $f:[0,1]tomathbb{R}$ is a continuous function in $[0,1]$ and differentiable in (0,1) such that $f(0)=0$ and $f(1)=1$. Prove that for all $kinmathbb{N}$ exist different $x_1,...,x_kin (0,1)$ such that $$sum_{i=1}^{k}frac{1}{f'(x_i)}=k$$
I'll be grateful for your help.
calculus derivatives continuity
edited Jan 7 at 1:11
Clayton
18.9k33085
asked Jan 7 at 0:57
Raul_MFerRaul_MFer
453
$endgroup$
I don't have any idea of how to proceed. The complete problem is:
If $f:[0,1]tomathbb{R}$ is a continuous function in $[0,1]$ and differentiable in (0,1) such that $f(0)=0$ and $f(1)=1$. Prove that for all $kinmathbb{N}$ exist different $x_1,...,x_kin (0,1)$ such that $$sum_{i=1}^{k}frac{1}{f'(x_i)}=k$$
I'll be grateful for your help.
calculus derivatives continuity
calculus derivatives continuity
edited Jan 7 at 1:11
Clayton
18.9k33085
asked Jan 7 at 0:57
Raul_MFerRaul_MFer
453
edited Jan 7 at 1:11
Clayton
18.9k33085
asked Jan 7 at 0:57
Raul_MFerRaul_MFer
453
edited Jan 7 at 1:11
Clayton
18.9k33085
edited Jan 7 at 1:11
Clayton
18.9k33085
edited Jan 7 at 1:11
Clayton
18.9k33085
18.9k33085
asked Jan 7 at 0:57
Raul_MFerRaul_MFer
453
asked Jan 7 at 0:57
Raul_MFerRaul_MFer
453
asked Jan 7 at 0:57
Raul_MFerRaul_MFer
453
453
put on hold as off-topic by user21820, José Carlos Santos, amWhy, Did, RRL Jan 7 at 17:34
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." – user21820, José Carlos Santos, amWhy, Did, RRL
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 user21820, José Carlos Santos, amWhy, Did, RRL Jan 7 at 17:34
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." – user21820, José Carlos Santos, amWhy, Did, RRL
If this question can be reworded to fit the rules in the help center, please edit the question.
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– yuanming luo
Jan 7 at 1:30
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– yuanming luo
Jan 7 at 1:30
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math.stackexchange.com/questions/366379/…
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– yuanming luo
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1 Answer
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Hint: there are $t_0 = 0 < ldots < t_k = 1$ such that $f(t_j) = j/k$. Use the Mean Value Theorem in each interval $[t_{j-1}, t_j]$.
answered Jan 7 at 1:13
Robert IsraelRobert Israel
319k23208458
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: there are $t_0 = 0 < ldots < t_k = 1$ such that $f(t_j) = j/k$. Use the Mean Value Theorem in each interval $[t_{j-1}, t_j]$.
answered Jan 7 at 1:13
Robert IsraelRobert Israel
319k23208458
$endgroup$
add a comment |
$begingroup$
Hint: there are $t_0 = 0 < ldots < t_k = 1$ such that $f(t_j) = j/k$. Use the Mean Value Theorem in each interval $[t_{j-1}, t_j]$.
answered Jan 7 at 1:13
Robert IsraelRobert Israel
319k23208458
$endgroup$
add a comment |
$begingroup$
Hint: there are $t_0 = 0 < ldots < t_k = 1$ such that $f(t_j) = j/k$. Use the Mean Value Theorem in each interval $[t_{j-1}, t_j]$.
answered Jan 7 at 1:13
Robert IsraelRobert Israel
319k23208458
$endgroup$
Hint: there are $t_0 = 0 < ldots < t_k = 1$ such that $f(t_j) = j/k$. Use the Mean Value Theorem in each interval $[t_{j-1}, t_j]$.
answered Jan 7 at 1:13
Robert IsraelRobert Israel
319k23208458
answered Jan 7 at 1:13
Robert IsraelRobert Israel
319k23208458
answered Jan 7 at 1:13
Robert IsraelRobert Israel
319k23208458
answered Jan 7 at 1:13
Robert IsraelRobert Israel
319k23208458
319k23208458
add a comment |
add a comment |
2
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math.stackexchange.com/questions/366379/…
$endgroup$
– yuanming luo
Jan 7 at 1:30