If $f(x)≤x$ , then $f′(x)≤1$?Finding moment of inertia across a solid sphere?Where can I find introductory video lectures about calculus and analysis?Accuracy of linear approximationsif $lim a_n = infty$ and $lim b_n = B$, then $lim (a_n+b_n) = infty$Studying analysis 2 from Munkres and SpivakForming an algebraic expression that is equal to its linear approximationIs the following function differentiable at $x=0$?How to determine whether a trig integral is True or False?Trigonometric functions differentiation with number inside the argument part.Inequality of derivatives
Why would five hundred and five same as one?
Is there any common country to visit for persons holding UK and Schengen visas?
In the event of Brexit being postponed beyond the EU elections, will UK voters in EU countries be eligible to participate?
Travelling in US for more than 90 days
Would this string work as string?
Why is participating in the European Parliamentary elections used as a threat?
Why didn't Voldemort know what Grindelwald looked like?
How do I lift the insulation blower into the attic?
How can a new country break out from a developed country without war?
Friend wants my recommendation but I don't want to give it to him
Magnifying glass in hyperbolic space
Output visual diagram of picture
How do you say "Trust your struggle." in French?
How to test the sharpness of a knife?
Can creatures abilities target that creature itself?
C++ lambda syntax
Error in master's thesis, I do not know what to do
Weird lines in Microsoft Word
What properties make a magic weapon befit a Rogue more than a DEX-based Fighter?
How would a solely written language work mechanically
Make a Bowl of Alphabet Soup
Why is indicated airspeed rather than ground speed used during the takeoff roll?
Reason why a kingside attack is not justified
Calculate Pi using Monte Carlo
If $f(x)≤x$ , then $f′(x)≤1$?
Finding moment of inertia across a solid sphere?Where can I find introductory video lectures about calculus and analysis?Accuracy of linear approximationsif $lim a_n = infty$ and $lim b_n = B$, then $lim (a_n+b_n) = infty$Studying analysis 2 from Munkres and SpivakForming an algebraic expression that is equal to its linear approximationIs the following function differentiable at $x=0$?How to determine whether a trig integral is True or False?Trigonometric functions differentiation with number inside the argument part.Inequality of derivatives
$begingroup$
I'm studying Calculus and having a trouble solving this question.
1) If $f(x)leq x$, then $f′(x)leq 1$ for all $x$?
2) What if $f(0)=0$, $f′(x)$ exists for all $x$?
I could easily find the counter example for 1) (Therefore it is false)
But I'm not sure about 2)
If $f(0)=0$ and $f′(x)$ exists for all $x$ &
$f(x)leq x$ , then $f′(x)leq 1$ for all $x$?
Please leave a comment if you don't mind :)
calculus derivatives
New contributor
$endgroup$
add a comment |
$begingroup$
I'm studying Calculus and having a trouble solving this question.
1) If $f(x)leq x$, then $f′(x)leq 1$ for all $x$?
2) What if $f(0)=0$, $f′(x)$ exists for all $x$?
I could easily find the counter example for 1) (Therefore it is false)
But I'm not sure about 2)
If $f(0)=0$ and $f′(x)$ exists for all $x$ &
$f(x)leq x$ , then $f′(x)leq 1$ for all $x$?
Please leave a comment if you don't mind :)
calculus derivatives
New contributor
$endgroup$
$begingroup$
The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
$endgroup$
– b00n heT
2 days ago
1
$begingroup$
I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
$endgroup$
– littleO
2 days ago
1
$begingroup$
Don't remove relevant information from your question!
$endgroup$
– user21820
2 days ago
add a comment |
$begingroup$
I'm studying Calculus and having a trouble solving this question.
1) If $f(x)leq x$, then $f′(x)leq 1$ for all $x$?
2) What if $f(0)=0$, $f′(x)$ exists for all $x$?
I could easily find the counter example for 1) (Therefore it is false)
But I'm not sure about 2)
If $f(0)=0$ and $f′(x)$ exists for all $x$ &
$f(x)leq x$ , then $f′(x)leq 1$ for all $x$?
Please leave a comment if you don't mind :)
calculus derivatives
New contributor
$endgroup$
I'm studying Calculus and having a trouble solving this question.
1) If $f(x)leq x$, then $f′(x)leq 1$ for all $x$?
2) What if $f(0)=0$, $f′(x)$ exists for all $x$?
I could easily find the counter example for 1) (Therefore it is false)
But I'm not sure about 2)
If $f(0)=0$ and $f′(x)$ exists for all $x$ &
$f(x)leq x$ , then $f′(x)leq 1$ for all $x$?
Please leave a comment if you don't mind :)
calculus derivatives
calculus derivatives
New contributor
New contributor
edited 2 days ago
user21820
39.6k543156
39.6k543156
New contributor
asked 2 days ago
Mighty QWERTYMighty QWERTY
325
325
New contributor
New contributor
$begingroup$
The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
$endgroup$
– b00n heT
2 days ago
1
$begingroup$
I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
$endgroup$
– littleO
2 days ago
1
$begingroup$
Don't remove relevant information from your question!
$endgroup$
– user21820
2 days ago
add a comment |
$begingroup$
The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
$endgroup$
– b00n heT
2 days ago
1
$begingroup$
I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
$endgroup$
– littleO
2 days ago
1
$begingroup$
Don't remove relevant information from your question!
$endgroup$
– user21820
2 days ago
$begingroup$
The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
$endgroup$
– b00n heT
2 days ago
$begingroup$
The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
$endgroup$
– b00n heT
2 days ago
1
1
$begingroup$
I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
$endgroup$
– littleO
2 days ago
$begingroup$
I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
$endgroup$
– littleO
2 days ago
1
1
$begingroup$
Don't remove relevant information from your question!
$endgroup$
– user21820
2 days ago
$begingroup$
Don't remove relevant information from your question!
$endgroup$
– user21820
2 days ago
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Hint: Consider $$f(x)=x-A sin^2 x $$
for large $A$.
$endgroup$
add a comment |
$begingroup$
Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^-x$ has $f'(x)gt1$ for all $x$, while $f(x)=x-1over2x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.
Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^-x$, of course, does not satisfy that condition.
$endgroup$
add a comment |
$begingroup$
Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.
New contributor
$endgroup$
$begingroup$
$f(x)leq x$ fails for $x$ negative.
$endgroup$
– Wojowu
2 days ago
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Mighty QWERTY is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3151380%2fif-fx%25e2%2589%25a4x-then-f-x%25e2%2589%25a41%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: Consider $$f(x)=x-A sin^2 x $$
for large $A$.
$endgroup$
add a comment |
$begingroup$
Hint: Consider $$f(x)=x-A sin^2 x $$
for large $A$.
$endgroup$
add a comment |
$begingroup$
Hint: Consider $$f(x)=x-A sin^2 x $$
for large $A$.
$endgroup$
Hint: Consider $$f(x)=x-A sin^2 x $$
for large $A$.
answered 2 days ago
user1337user1337
16.8k43592
16.8k43592
add a comment |
add a comment |
$begingroup$
Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^-x$ has $f'(x)gt1$ for all $x$, while $f(x)=x-1over2x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.
Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^-x$, of course, does not satisfy that condition.
$endgroup$
add a comment |
$begingroup$
Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^-x$ has $f'(x)gt1$ for all $x$, while $f(x)=x-1over2x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.
Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^-x$, of course, does not satisfy that condition.
$endgroup$
add a comment |
$begingroup$
Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^-x$ has $f'(x)gt1$ for all $x$, while $f(x)=x-1over2x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.
Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^-x$, of course, does not satisfy that condition.
$endgroup$
Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^-x$ has $f'(x)gt1$ for all $x$, while $f(x)=x-1over2x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.
Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^-x$, of course, does not satisfy that condition.
edited 2 days ago
answered 2 days ago
Barry CipraBarry Cipra
60.3k654126
60.3k654126
add a comment |
add a comment |
$begingroup$
Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.
New contributor
$endgroup$
$begingroup$
$f(x)leq x$ fails for $x$ negative.
$endgroup$
– Wojowu
2 days ago
add a comment |
$begingroup$
Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.
New contributor
$endgroup$
$begingroup$
$f(x)leq x$ fails for $x$ negative.
$endgroup$
– Wojowu
2 days ago
add a comment |
$begingroup$
Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.
New contributor
$endgroup$
Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.
New contributor
edited 2 days ago
Max
9011318
9011318
New contributor
answered 2 days ago
Alex KovalevskyAlex Kovalevsky
11
11
New contributor
New contributor
$begingroup$
$f(x)leq x$ fails for $x$ negative.
$endgroup$
– Wojowu
2 days ago
add a comment |
$begingroup$
$f(x)leq x$ fails for $x$ negative.
$endgroup$
– Wojowu
2 days ago
$begingroup$
$f(x)leq x$ fails for $x$ negative.
$endgroup$
– Wojowu
2 days ago
$begingroup$
$f(x)leq x$ fails for $x$ negative.
$endgroup$
– Wojowu
2 days ago
add a comment |
Mighty QWERTY is a new contributor. Be nice, and check out our Code of Conduct.
Mighty QWERTY is a new contributor. Be nice, and check out our Code of Conduct.
Mighty QWERTY is a new contributor. Be nice, and check out our Code of Conduct.
Mighty QWERTY is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3151380%2fif-fx%25e2%2589%25a4x-then-f-x%25e2%2589%25a41%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
$endgroup$
– b00n heT
2 days ago
1
$begingroup$
I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
$endgroup$
– littleO
2 days ago
1
$begingroup$
Don't remove relevant information from your question!
$endgroup$
– user21820
2 days ago