Question

How to solve this...
given f(x+2)-f(x) = (6x+4)^2 and f(0)=-16.
Find the value of f(5) ?

Answer 1

any ideas ?

Answer 2

Consider that you have a way to find f(2) since you know what f(0) is.

Answer 3

i got f(2) = 0

Answer 4

How?

Answer 5

i put x=0,
f(2) - f(0) = 4^2
f(2) - (-16) = 16
f(2) = 0 right ?

Answer 6

what;s next ? :)

Answer 7

i confused how going to f(5) ?

Answer 8

Is there any mention that f(x) is a cubic function? even if it is and assuming that the problem is true for any function satisfying the given condition, then f(5)=564

Answer 9

even if it is NOT I meant. Notice that \(f(x)=6x^3-6x^2-4x-16\) satisfying the given condition.

Answer 10

hmmmm... didnt say if f(x) is a cubic function

Answer 11

OK, I think I have an idea. But it needs an assumption that \(f\) is differentiable three times. Define \(h(x)=f(x+2)-f(x)\). Show that \(h'''(x)\) is constant. From there I think we can conclude that \(f(x)\) is a cubic function

Answer 12

What kind of magic did you do to come up with that cubic equation in the first place? I'm dumbfounded on two accounts of this problem lol.

Answer 13

i get the new problem, how to differentiate it ? :)

Answer 14

I don't think it is possible to find f(x) if x is odd number with the given information.

Answer 15

^ yeah we can find f(4) and f(6), but idk how to find f(5)...

Answer 16

i verified what @watchmath said,,the eqn fits in!
how did you get that ?

Answer 17

At least using f(4) and f(6) we can get an estimate of where the value of f(5) should lie.

Answer 18

You could try plotting all the even points and then find a line that fits. I have honestly no clue how he got that either.

Answer 19

Where did this problem come from anyways?

Answer 20

if begins with the assumption that \(f(x)\) is a cubic polynomial, and then curve-fitting using 4 different points.

Answer 21

this is my friend's problem... but i cant help him

Answer 22

What class is your friend in?

Answer 23

grade 12

Answer 24

I feel like there's a simple way to solve this that we just didn't notice.

Answer 25

i think so, but i dont know by what ?

Answer 26

This a second order difference equation with only one initial condition so the general solution is
\[
f(x)=-a (-1)^x+a+6 x^3-6 x^2-4 x-18 (-1)^x+2
\]
Where a is a constant.
f(5) =600 +2 a
where a can be anything

Answer 27

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