Quadratic problem. any other method?

[PrashntS]

quadratic problem. any other method??

1. Thie problem statement, all variables and given/known data
I was solving this question paper and came across the second problem :
http://olympiads.hbcse.tifr.res.in/uploads/rmo-2010

I wanted to solve this one by calculus.

I want to know that, does in problem like this, we may use:
If P(x)=Q(x)
Then, P'(x)=Q'(x), P''(x)=Q''(x) etc.?

I know this equality may not be true for other functions, but is it applicable for this specific problem?

 

[ehild]

You can not assume that the functions are identical, they just happen to return the same value at a specific x.
See the picture: although all function have the same value at a certain x, their derivatives are different.

That is an other thing that these special functions can have point(s) in common only when they are identical.

ehild

 

[PrashntS]

Thanks for reply. Yes I understand what you meant. Can you suggest some good method for this problem?
Let P1(x) = ax² − bx − c,
P2(x) = bx² − cx − a,
P3(x) = cx² − ax − b
be three quadratic
polynomials where a, b, c are non-zero real numbers. Suppose there exists a real number
α such that P1(α) = P2(α) = P3(α). Prove that a = b = c.

 

[ehild]

Sorry, I can not find any other method different from that given. It is a nice and simple method, (except a mistake in sign, but it does not matter) what is your problem with it?

ehild