Mixing
Continuous functions show that a function is decreasing
$begingroup$
Let $f:[0,infty)to[0,infty)$ be a continuously twice-differentiable, strictly increasing, and concave (also called concave down; i.e. $f''< 0)$ function such that $f(0)=0$.
A: Show that the following function $varphi:[0,infty)tomathbb{R}$ is decreasing for any fixed $t>0$:
$$varphi(x)=frac{f(x+t)−f(x)}{t}.$$
Will someone walk me through this problem, please? I do not understand how I prove this problem without any numbers. I also do not understand how to prove this for any $t>0$. Because just by giving an example say $t=14$, that proves the function for $t=14$. How do I prove it for the rest?
calculus
edited Feb 1 at 1:09
zipirovich
11.4k11731
asked Feb 1 at 0:36
DataD96DataD96
295
$endgroup$
add a comment |
$begingroup$
Let $f:[0,infty)to[0,infty)$ be a continuously twice-differentiable, strictly increasing, and concave (also called concave down; i.e. $f''< 0)$ function such that $f(0)=0$.
A: Show that the following function $varphi:[0,infty)tomathbb{R}$ is decreasing for any fixed $t>0$:
$$varphi(x)=frac{f(x+t)−f(x)}{t}.$$
Will someone walk me through this problem, please? I do not understand how I prove this problem without any numbers. I also do not understand how to prove this for any $t>0$. Because just by giving an example say $t=14$, that proves the function for $t=14$. How do I prove it for the rest?
calculus
edited Feb 1 at 1:09
zipirovich
11.4k11731
asked Feb 1 at 0:36
DataD96DataD96
295
$endgroup$
2
$begingroup$
This has nothing to do with set theory.
$endgroup$
– Noah Schweber
Feb 1 at 0:44
$begingroup$
Well, its a homework problem in my set theory class. Which is why I am confused.
$endgroup$
– DataD96
Feb 1 at 0:46
add a comment |
$begingroup$
Let $f:[0,infty)to[0,infty)$ be a continuously twice-differentiable, strictly increasing, and concave (also called concave down; i.e. $f''< 0)$ function such that $f(0)=0$.
A: Show that the following function $varphi:[0,infty)tomathbb{R}$ is decreasing for any fixed $t>0$:
$$varphi(x)=frac{f(x+t)−f(x)}{t}.$$
Will someone walk me through this problem, please? I do not understand how I prove this problem without any numbers. I also do not understand how to prove this for any $t>0$. Because just by giving an example say $t=14$, that proves the function for $t=14$. How do I prove it for the rest?
calculus
edited Feb 1 at 1:09
zipirovich
11.4k11731
asked Feb 1 at 0:36
DataD96DataD96
295
$endgroup$
Let $f:[0,infty)to[0,infty)$ be a continuously twice-differentiable, strictly increasing, and concave (also called concave down; i.e. $f''< 0)$ function such that $f(0)=0$.
A: Show that the following function $varphi:[0,infty)tomathbb{R}$ is decreasing for any fixed $t>0$:
$$varphi(x)=frac{f(x+t)−f(x)}{t}.$$
Will someone walk me through this problem, please? I do not understand how I prove this problem without any numbers. I also do not understand how to prove this for any $t>0$. Because just by giving an example say $t=14$, that proves the function for $t=14$. How do I prove it for the rest?
calculus
calculus
edited Feb 1 at 1:09
zipirovich
11.4k11731
asked Feb 1 at 0:36
DataD96DataD96
295
edited Feb 1 at 1:09
zipirovich
11.4k11731
asked Feb 1 at 0:36
DataD96DataD96
295
edited Feb 1 at 1:09
zipirovich
11.4k11731
edited Feb 1 at 1:09
zipirovich
11.4k11731
edited Feb 1 at 1:09
zipirovich
11.4k11731
11.4k11731
asked Feb 1 at 0:36
DataD96DataD96
295
asked Feb 1 at 0:36
DataD96DataD96
295
asked Feb 1 at 0:36
DataD96DataD96
295
295
2
$begingroup$
This has nothing to do with set theory.
$endgroup$
– Noah Schweber
Feb 1 at 0:44
$begingroup$
Well, its a homework problem in my set theory class. Which is why I am confused.
$endgroup$
– DataD96
Feb 1 at 0:46
add a comment |
2
$begingroup$
This has nothing to do with set theory.
$endgroup$
– Noah Schweber
Feb 1 at 0:44
$begingroup$
Well, its a homework problem in my set theory class. Which is why I am confused.
$endgroup$
– DataD96
Feb 1 at 0:46
2
2
$begingroup$
This has nothing to do with set theory.
$endgroup$
– Noah Schweber
Feb 1 at 0:44
$begingroup$
This has nothing to do with set theory.
$endgroup$
– Noah Schweber
Feb 1 at 0:44
$begingroup$
Well, its a homework problem in my set theory class. Which is why I am confused.
$endgroup$
– DataD96
Feb 1 at 0:46
$begingroup$
Well, its a homework problem in my set theory class. Which is why I am confused.
$endgroup$
– DataD96
Feb 1 at 0:46
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Just take the derivative of this new function. Note that from the point of view of this question, the only variable in it is $x$, while $t$ is treated as a parameter (i.e. as a constant when taking derivatives). So:
$$varphi'(x)=frac{f'(x+t)−f'(x)}{t}.$$
Now note that since the given function $f$ is concave, i.e. $f''<0$, we know that the first derivative $f'$ is decreasing. What does that tell us about the numerator above?
answered Feb 1 at 1:12
zipirovichzipirovich
11.4k11731
$endgroup$
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
});
}
});
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%2f3095678%2fcontinuous-functions-show-that-a-function-is-decreasing%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Just take the derivative of this new function. Note that from the point of view of this question, the only variable in it is $x$, while $t$ is treated as a parameter (i.e. as a constant when taking derivatives). So:
$$varphi'(x)=frac{f'(x+t)−f'(x)}{t}.$$
Now note that since the given function $f$ is concave, i.e. $f''<0$, we know that the first derivative $f'$ is decreasing. What does that tell us about the numerator above?
answered Feb 1 at 1:12
zipirovichzipirovich
11.4k11731
$endgroup$
add a comment |
$begingroup$
Just take the derivative of this new function. Note that from the point of view of this question, the only variable in it is $x$, while $t$ is treated as a parameter (i.e. as a constant when taking derivatives). So:
$$varphi'(x)=frac{f'(x+t)−f'(x)}{t}.$$
Now note that since the given function $f$ is concave, i.e. $f''<0$, we know that the first derivative $f'$ is decreasing. What does that tell us about the numerator above?
answered Feb 1 at 1:12
zipirovichzipirovich
11.4k11731
$endgroup$
add a comment |
$begingroup$
Just take the derivative of this new function. Note that from the point of view of this question, the only variable in it is $x$, while $t$ is treated as a parameter (i.e. as a constant when taking derivatives). So:
$$varphi'(x)=frac{f'(x+t)−f'(x)}{t}.$$
Now note that since the given function $f$ is concave, i.e. $f''<0$, we know that the first derivative $f'$ is decreasing. What does that tell us about the numerator above?
answered Feb 1 at 1:12
zipirovichzipirovich
11.4k11731
$endgroup$
Just take the derivative of this new function. Note that from the point of view of this question, the only variable in it is $x$, while $t$ is treated as a parameter (i.e. as a constant when taking derivatives). So:
$$varphi'(x)=frac{f'(x+t)−f'(x)}{t}.$$
Now note that since the given function $f$ is concave, i.e. $f''<0$, we know that the first derivative $f'$ is decreasing. What does that tell us about the numerator above?
answered Feb 1 at 1:12
zipirovichzipirovich
11.4k11731
edited Feb 1 at 1:19
answered Feb 1 at 1:12
zipirovichzipirovich
11.4k11731
answered Feb 1 at 1:12
zipirovichzipirovich
11.4k11731
answered Feb 1 at 1:12
zipirovichzipirovich
11.4k11731
11.4k11731
add a comment |
add a comment |
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%2f3095678%2fcontinuous-functions-show-that-a-function-is-decreasing%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
2
$begingroup$
This has nothing to do with set theory.
$endgroup$
– Noah Schweber
Feb 1 at 0:44
$begingroup$
Well, its a homework problem in my set theory class. Which is why I am confused.
$endgroup$
– DataD96
Feb 1 at 0:46