How to prove that $f(x)=ax^2+bx+c$? [on hold]

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Suppose that $fin C^2(-infty,+infty)$, and

$$f(x+h)-f(x)=hf'(x+dfrach2), forall x, hinmathbfR,$$

Prove that $f(x)=ax^2+bx+c$.

calculus

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edited 2 days ago

Bernard

110k635102

asked 2 days ago

闫嘉琦

32218

put on hold as off-topic by José Carlos Santos, Michael Hoppe, TheSimpliFire, Taroccoesbrocco, Brevan Ellefsen 2 days ago

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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Michael Hoppe, TheSimpliFire, Taroccoesbrocco, Brevan Ellefsen

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up vote
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Suppose that $fin C^2(-infty,+infty)$, and

$$f(x+h)-f(x)=hf'(x+dfrach2), forall x, hinmathbfR,$$

Prove that $f(x)=ax^2+bx+c$.

calculus

share|cite|improve this question

edited 2 days ago

Bernard

110k635102

asked 2 days ago

闫嘉琦

32218

put on hold as off-topic by José Carlos Santos, Michael Hoppe, TheSimpliFire, Taroccoesbrocco, Brevan Ellefsen 2 days ago

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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Michael Hoppe, TheSimpliFire, Taroccoesbrocco, Brevan Ellefsen

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Suppose that $fin C^2(-infty,+infty)$, and

$$f(x+h)-f(x)=hf'(x+dfrach2), forall x, hinmathbfR,$$

Prove that $f(x)=ax^2+bx+c$.

calculus

share|cite|improve this question

edited 2 days ago

Bernard

110k635102

asked 2 days ago

闫嘉琦

32218

Suppose that $fin C^2(-infty,+infty)$, and

$$f(x+h)-f(x)=hf'(x+dfrach2), forall x, hinmathbfR,$$

Prove that $f(x)=ax^2+bx+c$.

calculus

share|cite|improve this question

edited 2 days ago

Bernard

110k635102

asked 2 days ago

闫嘉琦

32218

share|cite|improve this question

share|cite|improve this question

edited 2 days ago

Bernard

110k635102

edited 2 days ago

Bernard

110k635102

edited 2 days ago

Bernard

110k635102

110k635102

asked 2 days ago

闫嘉琦

32218

asked 2 days ago

闫嘉琦

32218

asked 2 days ago

闫嘉琦

32218

32218

put on hold as off-topic by José Carlos Santos, Michael Hoppe, TheSimpliFire, Taroccoesbrocco, Brevan Ellefsen 2 days ago

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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Michael Hoppe, TheSimpliFire, Taroccoesbrocco, Brevan Ellefsen

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 José Carlos Santos, Michael Hoppe, TheSimpliFire, Taroccoesbrocco, Brevan Ellefsen 2 days ago

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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Michael Hoppe, TheSimpliFire, Taroccoesbrocco, Brevan Ellefsen

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

add a comment | 

1 Answer
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Only for harmony, lets replace $x$ by $x-h/2$ and then $h$ by $2h$. We get $$f(x+h)-f(x-h)=2hf'(x)$$

Taking derivatives with respect to $x$ $$f'(x+h)-f'(x-h)=2hf''(x)$$

Taking derivatives, of the first equation, with respect to $h$ $$f'(x+h)+f'(x-h)=2f'(x)$$

Adding $$2f'(x+h)=2hf''(x)+2f'(x)$$

Putting $x=0$ $$f'(h)=hf''(0)+f'(0)$$ Integrating you get your result.

share|cite|improve this answer

edited 2 days ago

Oscar Lanzi

9,75611631

answered 2 days ago

spiralstotheleft

30516

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote

Only for harmony, lets replace $x$ by $x-h/2$ and then $h$ by $2h$. We get $$f(x+h)-f(x-h)=2hf'(x)$$

Taking derivatives with respect to $x$ $$f'(x+h)-f'(x-h)=2hf''(x)$$

Taking derivatives, of the first equation, with respect to $h$ $$f'(x+h)+f'(x-h)=2f'(x)$$

Adding $$2f'(x+h)=2hf''(x)+2f'(x)$$

Putting $x=0$ $$f'(h)=hf''(0)+f'(0)$$ Integrating you get your result.

share|cite|improve this answer

edited 2 days ago

Oscar Lanzi

9,75611631

answered 2 days ago

spiralstotheleft

30516

add a comment | 

up vote
3
down vote

Only for harmony, lets replace $x$ by $x-h/2$ and then $h$ by $2h$. We get $$f(x+h)-f(x-h)=2hf'(x)$$

Taking derivatives with respect to $x$ $$f'(x+h)-f'(x-h)=2hf''(x)$$

Taking derivatives, of the first equation, with respect to $h$ $$f'(x+h)+f'(x-h)=2f'(x)$$

Adding $$2f'(x+h)=2hf''(x)+2f'(x)$$

Putting $x=0$ $$f'(h)=hf''(0)+f'(0)$$ Integrating you get your result.

share|cite|improve this answer

edited 2 days ago

Oscar Lanzi

9,75611631

answered 2 days ago

spiralstotheleft

30516

add a comment | 

up vote
3
down vote

up vote
3
down vote

Only for harmony, lets replace $x$ by $x-h/2$ and then $h$ by $2h$. We get $$f(x+h)-f(x-h)=2hf'(x)$$

Taking derivatives with respect to $x$ $$f'(x+h)-f'(x-h)=2hf''(x)$$

Taking derivatives, of the first equation, with respect to $h$ $$f'(x+h)+f'(x-h)=2f'(x)$$

Adding $$2f'(x+h)=2hf''(x)+2f'(x)$$

Putting $x=0$ $$f'(h)=hf''(0)+f'(0)$$ Integrating you get your result.

share|cite|improve this answer

edited 2 days ago

Oscar Lanzi

9,75611631

answered 2 days ago

spiralstotheleft

30516

Only for harmony, lets replace $x$ by $x-h/2$ and then $h$ by $2h$. We get $$f(x+h)-f(x-h)=2hf'(x)$$

Taking derivatives with respect to $x$ $$f'(x+h)-f'(x-h)=2hf''(x)$$

Taking derivatives, of the first equation, with respect to $h$ $$f'(x+h)+f'(x-h)=2f'(x)$$

Adding $$2f'(x+h)=2hf''(x)+2f'(x)$$

Putting $x=0$ $$f'(h)=hf''(0)+f'(0)$$ Integrating you get your result.

share|cite|improve this answer

edited 2 days ago

Oscar Lanzi

9,75611631

answered 2 days ago

spiralstotheleft

30516

share|cite|improve this answer

share|cite|improve this answer

edited 2 days ago

Oscar Lanzi

9,75611631

edited 2 days ago

Oscar Lanzi

9,75611631

edited 2 days ago

Oscar Lanzi

9,75611631

9,75611631

answered 2 days ago

spiralstotheleft

30516

answered 2 days ago

spiralstotheleft

30516

answered 2 days ago

spiralstotheleft

30516

30516

add a comment | 

add a comment |