Question

let f be a real fn satisfying f(x+y+z)=f(x)f(y)f(z) for all real x ,y ,zif f(2)=4 and f'(0)=3 find f(0) and f'(2). @Physics

Answer 1

james help

Answer 2

is that your pic btw?

Answer 3

:)

Answer 4

hw z it?

Answer 5

smoking

Answer 6

lol help me in this

Answer 7

.........................james?

Answer 8

Ok. Thinking about this a bit.

Answer 9

.........

Answer 10

Well, we need to evaluate f and f' and 0 and 2 obviously. What's not obvious to me right now is how to relate these quantities at x = 2.

But for x = 0, it's not so bad.

For example, f(3x) = f(x)^3 and hence 3 f'(3x) = 3 f'(x) f(x)^2 or
f'(3x) = f'(x) f(x)^2

Hence f'(0) = f'(0) f(0)^2 => f(0)^2 = 1 => f(0) = 1 or -1

Answer 11

othr one?

Answer 12

.,..

Answer 13

Let z = 0 and x = y = 1. Then
f(2) = f(1)^2 . f(0)

Also, if we let x = y, with z = 0, then
f(2x) = f(x)^2 . f(0)
hence
2 f'(2x) = 2 f(0) . f'(x) . f(x) ------ (*)

Again, not yet clear to me yet how to pull f'(2) from these.

Answer 14

hey i didnt get how u found f(0)

Answer 15

You agree that: f'(3x) = f'(x) f(x)^2 ?

Answer 16

how?

Answer 17

By differentiating both sides of f(3x) = f(x)^3

Answer 18

:( i didnt get whhere is y and z

Answer 19

Alternatively with that equation: f(3x) = f(x)^3
set x = 0 then f(0) = f(0)^3 => f(0) = -1, 0, 1
**
I set x = y = z hence
f(x + y + z) = f(3x)
and f(x)f(y)f(z) = f(x)^3

Answer 20

can i do that?

Answer 21

Sure; why not. x, y and z are arbitrary.

I have to go and have breakfast with friends. I'll think about the other part of this problem and post what I get.

Answer 22

when???

Answer 23

As soon as I have the answer. It bugs me when I don't have the answer, so I won't be able to fully stop thinking about it.

Answer 24

wel can u be online within 15 minutes?

Answer 25

I'll try, but I don't promise.

Answer 26

:( plzzzzz

Answer 27

Let z = 0 and x = y = 1. Then
f(2) = f(1)^2 . f(0)
we got f(0)=+1 or -1
and we know f(2)=4
hence
4=f(1)^2 *1 or 4=f(1)*-1
as f(1)^2 > 0
hence f(0)=+1

Answer 28

helpppppppppp

Answer 29

@sriram: that makes sense.

Answer 30

2nd part

Answer 31

Can you confirm that "f(2)=4 and f'(0)=3" is what the problem says?

Answer 32

no

Answer 33

So if it doesn't say that f(2) = 4 and f'(0) = 3, what does it say?

Answer 34

The reason I ask is that you'd think this function f is something like
\[ f(x) = e^{ax} \]
But with those values, I can't make that work.

However even setting that form of f aside, I'm running into other difficulties trying to solve this with those values.