Suppose that $f(x)$ is differentiable at any point and $f(0) = 0$ and $|f′(x)|leq 1$ . Prove that $|f(x)|...
Suppose that:
$f(x)$ is differentiable at any point- $f(0) = 0$
- $|f′(x)| ≤ 1$
How to prove $|f(x)| leq |x|$ for any $x$?
calculus
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Suppose that:
$f(x)$ is differentiable at any point- $f(0) = 0$
- $|f′(x)| ≤ 1$
How to prove $|f(x)| leq |x|$ for any $x$?
calculus
New contributor
put on hold as off-topic by Nosrati, Davide Giraudo, amWhy, mrtaurho, John Wayland Bales 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." – Nosrati, Davide Giraudo, amWhy, mrtaurho, John Wayland Bales
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
Suppose that:
$f(x)$ is differentiable at any point- $f(0) = 0$
- $|f′(x)| ≤ 1$
How to prove $|f(x)| leq |x|$ for any $x$?
calculus
New contributor
Suppose that:
$f(x)$ is differentiable at any point- $f(0) = 0$
- $|f′(x)| ≤ 1$
How to prove $|f(x)| leq |x|$ for any $x$?
calculus
calculus
New contributor
New contributor
edited 2 days ago
amWhy
192k28224439
192k28224439
New contributor
asked 2 days ago
Johnathan1994
1
1
New contributor
New contributor
put on hold as off-topic by Nosrati, Davide Giraudo, amWhy, mrtaurho, John Wayland Bales 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." – Nosrati, Davide Giraudo, amWhy, mrtaurho, John Wayland Bales
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 Nosrati, Davide Giraudo, amWhy, mrtaurho, John Wayland Bales 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." – Nosrati, Davide Giraudo, amWhy, mrtaurho, John Wayland Bales
If this question can be reworded to fit the rules in the help center, please edit the question.
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1 Answer
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Hint: By mean value theorem, there exists $xiinBbb R $ such that $$f'(xi)=frac{f (x)-f (0)}{x-0}=frac{f (x)}{x}$$. Now use the fact that $|f'(xi)|leq 1$.
1
Thanks a lot !!!
– Johnathan1994
2 days ago
Please read this before asking next question.
– Thomas Shelby
yesterday
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint: By mean value theorem, there exists $xiinBbb R $ such that $$f'(xi)=frac{f (x)-f (0)}{x-0}=frac{f (x)}{x}$$. Now use the fact that $|f'(xi)|leq 1$.
1
Thanks a lot !!!
– Johnathan1994
2 days ago
Please read this before asking next question.
– Thomas Shelby
yesterday
add a comment |
Hint: By mean value theorem, there exists $xiinBbb R $ such that $$f'(xi)=frac{f (x)-f (0)}{x-0}=frac{f (x)}{x}$$. Now use the fact that $|f'(xi)|leq 1$.
1
Thanks a lot !!!
– Johnathan1994
2 days ago
Please read this before asking next question.
– Thomas Shelby
yesterday
add a comment |
Hint: By mean value theorem, there exists $xiinBbb R $ such that $$f'(xi)=frac{f (x)-f (0)}{x-0}=frac{f (x)}{x}$$. Now use the fact that $|f'(xi)|leq 1$.
Hint: By mean value theorem, there exists $xiinBbb R $ such that $$f'(xi)=frac{f (x)-f (0)}{x-0}=frac{f (x)}{x}$$. Now use the fact that $|f'(xi)|leq 1$.
answered 2 days ago
Thomas Shelby
1,680216
1,680216
1
Thanks a lot !!!
– Johnathan1994
2 days ago
Please read this before asking next question.
– Thomas Shelby
yesterday
add a comment |
1
Thanks a lot !!!
– Johnathan1994
2 days ago
Please read this before asking next question.
– Thomas Shelby
yesterday
1
1
Thanks a lot !!!
– Johnathan1994
2 days ago
Thanks a lot !!!
– Johnathan1994
2 days ago
Please read this before asking next question.
– Thomas Shelby
yesterday
Please read this before asking next question.
– Thomas Shelby
yesterday
add a comment |