Jonask 2 years ago functional equations

1. Mertsj

love em or leave em

\[\color{brown}{f(x)+(x+1)^3=2f(x+1)}\]

3. ParthKohli

Find \(\color{#C00}{f(10)}\)

lol thanks for posting the full question @ParthKohli

5. ParthKohli

\[f(-1) = 2f(0)\]

f(0)+1=2f(1)

wat to do next i 3 variables 2 equations??

8. ParthKohli

\[f(0) = 2f(1) - 1 \\ f(1) = 2f(2) - 8\]So\[f(0) = 2(2f(2) - 8) = 4f(2) - 16\]

9. ParthKohli

So hard...

\[f(0)=4(f(2)-8)-1=4f(2)-16-1=4f(2)-17\]

is f always of the from\[ax^2+bx+c\]

12. ParthKohli

13. ParthKohli

No, it's a polynomial. It can be in that form, but we're not sure.

so far we see \[f(10)+11^3=2f(11)\] \[f(9)+10^3=2f(10)\] ---------------------------------------------- \[f(10)=f(9)-2f(11)+11^3-10^3\]

15. ParthKohli

I think that it must be in the form \(ax^3 + bx^2 + cx + d\)

\[f(x-1)+x^3=2f(x)\]

we need a relation between f(9) and f(11)

18. ParthKohli

\[f(9) = 2f(10) - 1000\]\[f(10) = 2f(11) - 1331\]\[\iff f(9) = 2(f(11) - 1331 ) - 1000\]

19. ParthKohli

\[f(9) = 2f(11) - 2662 - 1000\]

20. ParthKohli

That's the relation.

isnt this the solution then

since \[f(9)-2f(11)=3662\] \[f(10)=3662+11^3-10^3\]

23. ParthKohli

OMG!

24. ParthKohli

But it's an integer between 0 and 999.

okay so we found a relationship between numbers 2 units away from each other x-1 and x+1 \[\color{blue}{f(x-1)+x^3=2f(x).......... f(x)+(1+x)^3=2f(1+x)}\] \[f(x-1)+x^3=4f(x+1)+2(x+1)^3\]

\[f(9)-4f(11)=2(11)^3-10^3\]

earlier we used 2 not 4

28. ParthKohli

Ahhh.

29. ParthKohli

@Hero

30. sauravshakya

Let f(x)=a+bx+cx^2+dx^3 NOTE: f(x) must be a third degree polynomial

31. sauravshakya

Now, f(x)+(x+1)^3 =2f(x+1)

\[\huge f(10)=f(9)-2f(11)+11^3-10^3=???\]

33. sauravshakya

use it to find a,b,c and d

34. ParthKohli

I see, yeah!

35. sauravshakya

u will get a=-13 b=9 c=-3 d=1

36. ParthKohli

Equating coefficients

37. sauravshakya

yep

38. ParthKohli

Yay

39. sauravshakya

now, once u know f(x) u can find f(10)

help me i dont get where you get the coeeffients from

41. ParthKohli

@Jonask a technique.\[ax + by = 2x + 3y\]means\[a=2,b=3\]

oh great

43. sauravshakya

f(x)+(x+1)^3 =2f(x+1) a+bx+cx^2+dx^3 +x^3+3x^2+3x+1 = 2{a+b(x+1) +c(x+1)^2 +d(x+1)^3

44. sauravshakya

can u continue from here?

45. ParthKohli

\[f(10) = 10^3 - 3(10)^2 +9(10) -13 = 1000 - 300 + 90 - 13 \\ = 777 \]

46. sauravshakya

thats correct

\[\huge \color{green}{THANKS}\]

48. sauravshakya

welcome