Question

Help me please!!! Ayudame por favor!
Analysis proof.
Suppose that \(f:[a,b] \rightarrow Real \) is continuous. Suppose that \(\int_{a}^{x}f=\int_{x}^{b}f\) for every x is an element of [a,b]. Show f(x)=0 for every x is an element of [a,b].

Answer 1

I was thinking contradiction, but I just don't know where to start.

Answer 2

So if
ax∫f = F(x) - F(a)
xb∫f = F(b) - F(x)
I'm not done thinking but this is where I was starting.

Answer 3

ok so FTC to there makes sense

Answer 4

then just say if they are equal F(b) must = F(a) therefore no distance has been travelled?

Answer 5

so F(x)=0?

Answer 6

I was thinking of picking a point C in between them and plunging them back into the equation. So
F(x) - F(a) =F(b) - F(x)
becomes
F(x) - F(a) - [F(b) - F(x)] = 0

Answer 7

I'm sorry that doesn't get anywhere.

Answer 8

it's ok. I suck at this, I appreciate the effort... I'm wishing it was just calc again lol

Answer 9

What is this supposed to be? haha.

Answer 10

homework... :x

Answer 11

No I mean what math is this?

Answer 12

and it's for Real Analysis

Answer 13

pretty much prove calc 1,2,3 and parts of dif eq

Answer 14

What happens as x -> a?

Answer 15

The integral gets closer and closer to zero

Answer 16

If you take the derivative of both sides, you have
\[f(x)=-f(x)\]
which is only true for \(f(x)=0\), if I'm not mistaken. Something tells me there's more to this proof, though.

Answer 17

rhs

Answer 18

on the left and on the right it becomes the full integral from a to b of F(x) is equal to zero

Answer 19

thanks @SithsAndGiggles how did you get \(-f(x)\)?

Answer 20

x is on the bottom of the B integral therefore it would have been F(B) - F(x) then you would take the derivative.

Answer 21

Fundamental theorem of calc:
\[\frac{d}{dx}\int_c^{g(x)}f(t)~dt=f(g(x))\cdot g'(x)\]
So,
\[\frac{d}{dx}\int_x^bf(t)~dt=\frac{d}{dx}\left[-\int_b^xf(t)~dt\right]=-f(x)\]

Answer 22

hmm, I follow that @SithsAndGiggles I want to see how @pgpilot326 is doing this first because I have a feeling they are correct

Answer 23

yes pg agreed

Answer 24

then 2F(x)=0 and it can only happen when f(x) is zero?

Answer 25

and actually since it is for all x in [a,b], there is no need to take a limit.

Answer 26

The only way the two integrals could be equal for f(x) not equal to 0 is if there was a jump discontinuity.

Answer 27

@SithsAndGiggles : consider f(x) = e^-x
d/dx f(x) = -e^-x = -(e^-x)=-f(x)

Answer 28

if F(x) - F(a) = F(a) - F(x) then they are negatives of one another. the only number that satisfies this is 0. => f(x) = 0 for all x in [a,b].

Answer 29

Lets just assume you had a horizontal line with a constant-zero slope: C
Upon integration, you'd end up with a variable:
Which is untrue for all a not equal to b.