Polynomial of Degree 2

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January 19th 2011, 12:35 PM
zebra2147

Polynomial of Degree 2

My professor has this proof in his notes and I was wondering if anyone could help get me rolling...

Let $f :\mathbb{R} \rightarrow \mathbb{R}$ be differentiable. Suppose
$f(x + y) = f(x) + f(y) + 2xy$ for all $x,y\in \mathbb{R}$ .

Show by completing the two parts below that $f$ is a polynomial of degree 2.
(1) Prove $f'(x) = f'(0)+2x$ , for all $x\in \mathbb{R}$ .
(2) Prove $f(x) = x^2+f'(0)x+f(0)$ ,for all $x\in \mathbb{R}$ .

I think my biggest problem is coming up with the best way to represent $f'(x)$ which is obviously a big roadblock.
January 19th 2011, 12:44 PM
TheEmptySet

Quote:

Originally Posted by zebra2147

My professor has this proof in his notes and I was wondering if anyone could help get me rolling...

Let $f :\mathbb{R} \rightarrow \mathbb{R}$ be differentiable. Suppose
$f(x + y) = f(x) + f(y) + 2xy$ for all $x,y\in \mathbb{R}$ .

Show by completing the two parts below that $f$ is a polynomial of degree 2.
(1) Prove $f'(x) = f'(0)+2x$ , for all $x\in \mathbb{R}$ .
(2) Prove $f(x) = x^2+f'(0)x+f(0)$ ,for all $x\in \mathbb{R}$ .

I think my biggest problem is coming up with the best way to represent $f'(x)$ which is obviously a big roadblock.

If you take the derivative with respect to x you get

$\displaystyle f'(x+y)=f'(x)+2y \implies f'(x)=f'(x+y)-2y$

This must hold for all $y \in \mathbb{R}$ so what choice of $y$ will give the result that you want?
January 19th 2011, 01:01 PM
zebra2147

$y=-x$ ??

So that we get $f'(x) = f'(0)+2x$ ?
January 19th 2011, 01:23 PM
TheEmptySet

Quote:

Originally Posted by zebra2147

$y=-x$ ??

So that we get $f'(x) = f'(0)+2x$ ?

Yes that is the one:)
January 19th 2011, 01:53 PM
zebra2147

So is the information that you have helped me come up sufficient for the proof because it only has to work for all $x$ and not all $y$ ?

And do you have any hints for part 2?

Thanks!
January 19th 2011, 02:20 PM
Drexel28

Quote:

Originally Posted by zebra2147

So is the information that you have helped me come up sufficient for the proof because it only has to work for all $x$ and not all $y$ ?

And do you have any hints for part 2?

Thanks!

Let $g(x)=f(x)-x^2-f'(0)x$ . Note that $g'(x)=f'(x)-2x-f'(0)=0$ and since $g'(x)=0$ on an interval (in this case all of $\mathbb{R}$ ) we may conclude that $f(x)-x^2-f'(0)x=g(x)=c$ for some $c$ . Noting that $g(0)=f(0)$ gives the desired results.