Help: Functional Equation

If $f(x)$ is a polynomial in $x$ satisfying $f(x) f(y) + 2 = f(x) + f(y) + f(xy)$ and $f(0) = 0, f'(1) = 2$ , find $f(x)$ .

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The question is wrong . I will prove it . Put $x,y=0$ $f(0)^{2}-3f(0)+2=0$ This implies $f(0)=1,2$ now put $x,y=1$ this gives $f(1)=1,2$ Differentiating the function partially with respect to $x$ we get $f'(x)f(y)=f'(x)+yf'(xy).......(1)$ Now put $x,y=1$ in $(1)$ We get $f(1)=2$ Now put $x,y=0$ in $(1)$ We get $f(0)=1..or..f'(0)=0$ Now put $x=1$ in (1) we get $\frac{2f(y)-2}{y}=f'(y)...(2)$ Now let $f(0)=1$ , we have $f(1)=2$ then we observe that $f(x)=x+1$ also in satisfies main functional equation and (2),(1) but $f'(x)=1$ hence question is wrong . If only main functional equation is given without any condition then $f(x)=x+1$ .

Shivam Jadhav - 5 years, 2 months ago

Hey, if you see the other post, which is the same question, here,
you'll see we get,
$f(x) - 1 = x^t$
For no value of t, will both the given conditions be satisfied- there is no such function, as a result. Try it out! I think you should check your question again.

Ameya Daigavane - 5 years, 2 months ago

f(1)=2 and f'(0)=0 should be there for the question to be correct

parv mor - 5 years, 2 months ago

Ya.. I also reached the same conclusion...

Rishabh Jain - 5 years, 2 months ago

But your suggestion makes the functional equation pretty much useless, as for $x=1$ , and $y=c$ , $c\in dom(f)$ we do not get any relation to solve for $f(x)$

Akhilesh Prasad - 5 years, 2 months ago

Please if you can post a solution@parv mor

Akhilesh Prasad - 5 years, 2 months ago

I also came up with the same conclusion as you, i.e. $f(x)=2$ , so i thought of using the $f'(1)=2$ . So as we know from the definition of derivatives

$f'(1)=\frac { f(1+h)-f(1) }{ h }\\$

$\Longrightarrow f'(1)=\frac { f(1(1+h))-f(1) }{ h }$

From the functional equation we have, $\\$

$f(1)f(1+h))+2=f(1)+f(1+h)+f(1(1+h))$

$\Longrightarrow f(1+h)=f(1)f(1+h)-f(1)-f(1+h)+2$

$\therefore \quad f^{ 1 }\left( x \right) =\frac { f(1)f(1+h)-f(1)-f(1+h)+2-f(1) }{ h } =\frac { f(1)(f(1+h)-2)-(f(1+h)-2) }{ h }$

$\Longrightarrow f^{ 1 }\left( x \right) =\frac { (f(1)-1)(f(1+h)-2) }{ h }$

$\Longrightarrow 2h=(f(1)-1)(f(1+h)-2)$

From here on I am not able to conclude anything useful, so if you would help me lead in the right direction or tell me something that i am doing wrong.

Akhilesh Prasad - 5 years, 2 months ago

You have to take the limit for the derivative, as h goes to 0.

Ameya Daigavane - 5 years, 2 months ago

Is the question all right?? Put y=0 in the given functional equation to get: $0+2=f(x)+0+0~~~~\small{\text{(since f(0)=0)}}$ $\Large f(x)=2$ which does not satisfy the condition f'(1)=2 !!

(It might work if indeed it was given f(1)=2 and f'(0)=0)

Rishabh Jain - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$