Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+…+a_{1}x=0$ has positive roots, then...
Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.
P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience
derivatives
asked Nov 25 at 13:46
Trần Tuấn
254
add a comment |
Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.
P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience
derivatives
asked Nov 25 at 13:46
Trần Tuấn
254
Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– Martin R
Nov 25 at 13:50
1
This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
– Batominovski
Nov 25 at 13:51
Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
– xbh
Nov 25 at 13:51
add a comment |
Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.
P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience
derivatives
asked Nov 25 at 13:46
Trần Tuấn
254
Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.
P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience
derivatives
derivatives
asked Nov 25 at 13:46
Trần Tuấn
254
asked Nov 25 at 13:46
Trần Tuấn
254
asked Nov 25 at 13:46
Trần Tuấn
254
asked Nov 25 at 13:46
Trần Tuấn
254
asked Nov 25 at 13:46
Trần Tuấn
254
254
Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– Martin R
Nov 25 at 13:50
1
This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
– Batominovski
Nov 25 at 13:51
Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
– xbh
Nov 25 at 13:51
add a comment |
Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– Martin R
Nov 25 at 13:50
1
This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
– Batominovski
Nov 25 at 13:51
Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
– xbh
Nov 25 at 13:51
Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– Martin R
Nov 25 at 13:50
Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– Martin R
Nov 25 at 13:50
1
1
This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
– Batominovski
Nov 25 at 13:51
This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
– Batominovski
Nov 25 at 13:51
Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
– xbh
Nov 25 at 13:51
Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
– xbh
Nov 25 at 13:51
add a comment |
1 Answer
1
active
oldest
votes
Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
$$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
which is the required statement to be proven.
Hope it is helpful
answered Nov 25 at 13:56
Martund
1,384212
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%2f3012856%2fprove-that-if-a-nxna-n-1xn-1-a-1x-0-has-positive-roots-then%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
Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
$$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
which is the required statement to be proven.
Hope it is helpful
answered Nov 25 at 13:56
Martund
1,384212
add a comment |
Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
$$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
which is the required statement to be proven.
Hope it is helpful
answered Nov 25 at 13:56
Martund
1,384212
add a comment |
Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
$$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
which is the required statement to be proven.
Hope it is helpful
answered Nov 25 at 13:56
Martund
1,384212
Let $alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $mathbb R$)
$$Longrightarrow exists cin(0,alpha):f'(c) = 0$$
which is the required statement to be proven.
Hope it is helpful
answered Nov 25 at 13:56
Martund
1,384212
answered Nov 25 at 13:56
Martund
1,384212
answered Nov 25 at 13:56
Martund
1,384212
answered Nov 25 at 13:56
Martund
1,384212
1,384212
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3012856%2fprove-that-if-a-nxna-n-1xn-1-a-1x-0-has-positive-roots-then%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
Your question is clear to understand, but without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– Martin R
Nov 25 at 13:50
1
This is an easy application of Rolle's Theorem: en.wikipedia.org/wiki/Rolle%27s_theorem.
– Batominovski
Nov 25 at 13:51
Please add your thoughts about the question. The site discourages people asking question without even try to solve. No matter what attempts you made, type them into your post, and people would help you. Otherwise you would receive a lot of downvotes, and your post would highly likely be put on hold for further improvement.
– xbh
Nov 25 at 13:51