Question

Let \( f(x) \) and \( g(x) \) be two increasing and continuous functions on interval \( (a, b) \). Check weather the following integral is positive or not.

\[ \int_a^b f(x) g(x) dx \]

Also let

\[ \int_a^b f(x) g(x) dx =0 \]
define the scalar product of two functions \( f(x) \) and \( g(x) \)

Answer 1

@hidy give it a try.

Answer 2

okay

Answer 3

NEI; not enough information

Answer 4

f(x)g(x) can be below the x-axis or above the x-axis

Answer 5

f(x) and g(x) can be negative or positive

Answer 6

yes, but in the giving integral you choosed to work in one dimension IR
what if the integral is defined like this way :\[\int\limits\int\limits f(x)g(y)dxdy\]

Answer 7

remember that we want to proove it for the bilinear form B(x,y)

Answer 8

Sacred faeces this is getting intense

Answer 9

looks like I didn't not put up enough information!! ... this was to check the claim by @hidy
"if we compose a bilinear form with two increasing functions do we get a positive biliear form"

Answer 10

The integral could be positive or negative

Answer 11

f(x) = x^2 and g(x) = x - 1, on [0,1], the integral is negative

or

f(x) = x^2 and g(x) = x , on [0,1] the integral is positive

Answer 12

the biliner form dosen't necessary define a scalar product

Answer 13

I don't think we need to define the scalar product that way ... @hidy ... probably you would do that for smooth functions of two variables i.e. f(x,y), g(x,y) on some interval (a,b)x(c,d).
just stick with single integral for one variable. If there is counterexample then, your claim is invalid.

Answer 14

probably I should make the interval (a,b) such that
\[\int_a^b f(x) g(x) dx = 0 \]
if f(x) and f(x) are linearly independent.

Answer 15

well ... you are definitely right, but we need a map there that maps two functions into field.
couldn't think any better than scalar product.

Answer 16

and what can that give us ??

Answer 17

yes I see

Answer 18

i kinda feel hard to define linear independence ... to make sourwing claim invalid.

Answer 19

I got an idea but I don't know how to use it
we know that we can associate a quadratic form to a bilnear form that satisfies B(x,x)=q(x) and q is always positive

Answer 20

still you need a map
\[ B:L^2\times L^2 \to \Bbb R \]

Answer 21

yeah :(

Answer 22

http://en.wikipedia.org/wiki/Quadratic_form
the first example you see is for vector space (x,y) ... this x, and y represents numbers.

Answer 23

check this out
The associated bilinear form of a quadratic form q is defined by

\[ b_q(x,y)=\tfrac{1}{2}(q(x+y)-q(x)-q(y)) = x^\mathrm{T}Ay = y^\mathrm{T}Ax. \]

Answer 24

so obviously the bilinear form is not always positive
we should have the condition
q(x+y)>q(x)+q(y)

Answer 25

no it won't ...

Answer 26

so I guess that's it we cant' go farther but if I find something I'll show you
thanks for your help :)

Answer 27

yw ... i hope someone answers it, algebra isn't my cup of tea.

Answer 28

yw ... i hope someone answers it, algebra ain't my cup of tea.

Answer 29

no no it was an amazing effort I'm impressed :)