What is the underlying concept behind this non-existent function problem?

[sutupidmath]

Explain this a little please

well i have another problem with finding a(some) functions. The problem is this:
FInd all non-negative continuous functions f:[01]-->R that fullfil the following:

integ from 0 to 1 of f(x)dx =1

integ from 0 to 1 of xf(x)dx=a and

integ from 0 to 1 of x^2 f(x)dx=a^2

where a is an element of the reals.

this problem is in a textbook, and it starts like this

integ from 0 to one of (x-a)^2 f(x) dx, and at the end comes to a contradiction, hence it concludes that such a functions does not exist.

My question is what is really this problem asking?
SO if you could explain a little to me how should i start thinking about this problem? Because i want to fully understand what i am writing, and why am i writing something.
So could anyone please elaborate this problem a little for me?


I really thank all you guys for your time, and for taking the effort on helping us all.

P.S. Btw it is not homework, just a problem i encountered in the textbook.

 

[mathman]

Essentially, the int. from 0 to 1 of (x-a)^2f(x) =0, therefore f(x)=0, contradicting the original assumptions.

 

[sutupidmath]

Essentially, the int. from 0 to 1 of (x-a)^2f(x) =0, therefore f(x)=0, contradicting the original assumptions.

Yeah, as i stated i got to this contradiction. I can follow all these steps, but my point is that for example why we even started with the int. from 0 to 1 of (x-a)^2f(x) ? I just can't get the point of this problem, what does it really mean. Because i have done it, but more like in a mechanical way, not a logical one. That is why i want to know the logic behind it, or the reasoning?

For example, let us suppose that there were no instructions at all, then how would i have to think about this problem in order to start solving it. Because as long as i saw that instruction, i managed to do the rest, but like i stated, not in a reall logical way.

 

[sutupidmath]

so is anyone else going to give me some instructions?

 

[ZioX]

It's used because it produces a contradiction. It is a contrived question.

 

[sutupidmath]

It's used because it produces a contradiction. It is a contrived question.

Yeah, i have figured that out as well. Maybe i am being a little ambiguous. What i want to know is that if we were given just the problem

FInd all non-negative continuous functions f:[01]-->R that fullfil the following:

integ from 0 to 1 of f(x)dx =1

integ from 0 to 1 of xf(x)dx=a and

integ from 0 to 1 of x^2 f(x)dx=a^2

where a is an element of the reals.

and not any instruction at all, then how should i tackle this problem?
Assume that you have never seen this problem solved before, then what would you do?? How would one know at first place that such a function does not exist, or we should just suppose it. But again how to start?
I want to know how to get the idea of approching this problem, and similar ones?

 

[matt grime]

One wouldn't know such a function did or did not exist. One would try to find what kinds of things these conditions imply. One would think what things it reminds one of. Where do I see integrals of f, xf and x^2f? If I couldn't think of any such cases then I'd just play around - suppose that f has a taylor series and I can do term by term integration, what would that imply? I'd also think about variations on the problem.

What I personally would notices is:

it defines is a probability distribution on [0,1] with mean a and second moment a^2, but that implies that the variance is zero. That is clearly impossible, and in fact implies the hint: the variance is the integral of (x-a)^2f(x).
More abstractly what is really going on is that any such f would have to contradict something like the Cauchy Schwartz inequality.