anonymous
  • anonymous
functional equations
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
Mertsj
  • Mertsj
love em or leave em
anonymous
  • anonymous
\[\color{brown}{f(x)+(x+1)^3=2f(x+1)}\]
ParthKohli
  • ParthKohli
Find \(\color{#C00}{f(10)}\)

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anonymous
  • anonymous
lol thanks for posting the full question @ParthKohli
ParthKohli
  • ParthKohli
\[f(-1) = 2f(0)\]
anonymous
  • anonymous
f(0)+1=2f(1)
anonymous
  • anonymous
wat to do next i 3 variables 2 equations??
ParthKohli
  • ParthKohli
\[f(0) = 2f(1) - 1 \\ f(1) = 2f(2) - 8\]So\[f(0) = 2(2f(2) - 8) = 4f(2) - 16\]
ParthKohli
  • ParthKohli
So hard...
anonymous
  • anonymous
\[f(0)=4(f(2)-8)-1=4f(2)-16-1=4f(2)-17\]
anonymous
  • anonymous
is f always of the from\[ax^2+bx+c\]
ParthKohli
  • ParthKohli
Ah my bad, yes.
ParthKohli
  • ParthKohli
No, it's a polynomial. It can be in that form, but we're not sure.
anonymous
  • anonymous
so far we see \[f(10)+11^3=2f(11)\] \[f(9)+10^3=2f(10)\] ---------------------------------------------- \[f(10)=f(9)-2f(11)+11^3-10^3\]
ParthKohli
  • ParthKohli
I think that it must be in the form \(ax^3 + bx^2 + cx + d\)
anonymous
  • anonymous
\[f(x-1)+x^3=2f(x)\]
anonymous
  • anonymous
we need a relation between f(9) and f(11)
ParthKohli
  • ParthKohli
\[f(9) = 2f(10) - 1000\]\[f(10) = 2f(11) - 1331\]\[\iff f(9) = 2(f(11) - 1331 ) - 1000\]
ParthKohli
  • ParthKohli
\[f(9) = 2f(11) - 2662 - 1000\]
ParthKohli
  • ParthKohli
That's the relation.
anonymous
  • anonymous
isnt this the solution then
anonymous
  • anonymous
since \[f(9)-2f(11)=3662\] \[f(10)=3662+11^3-10^3\]
ParthKohli
  • ParthKohli
OMG!
ParthKohli
  • ParthKohli
But it's an integer between 0 and 999.
anonymous
  • anonymous
okay so we found a relationship between numbers 2 units away from each other x-1 and x+1 \[\color{blue}{f(x-1)+x^3=2f(x).......... f(x)+(1+x)^3=2f(1+x)}\] \[f(x-1)+x^3=4f(x+1)+2(x+1)^3\]
anonymous
  • anonymous
\[f(9)-4f(11)=2(11)^3-10^3\]
anonymous
  • anonymous
earlier we used 2 not 4
ParthKohli
  • ParthKohli
Ahhh.
ParthKohli
  • ParthKohli
@Hero
anonymous
  • anonymous
Let f(x)=a+bx+cx^2+dx^3 NOTE: f(x) must be a third degree polynomial
anonymous
  • anonymous
Now, f(x)+(x+1)^3 =2f(x+1)
anonymous
  • anonymous
\[\huge f(10)=f(9)-2f(11)+11^3-10^3=???\]
anonymous
  • anonymous
use it to find a,b,c and d
ParthKohli
  • ParthKohli
I see, yeah!
anonymous
  • anonymous
u will get a=-13 b=9 c=-3 d=1
ParthKohli
  • ParthKohli
Equating coefficients
anonymous
  • anonymous
yep
ParthKohli
  • ParthKohli
Yay
anonymous
  • anonymous
now, once u know f(x) u can find f(10)
anonymous
  • anonymous
help me i dont get where you get the coeeffients from
ParthKohli
  • ParthKohli
@Jonask a technique.\[ax + by = 2x + 3y\]means\[a=2,b=3\]
anonymous
  • anonymous
oh great
anonymous
  • anonymous
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
anonymous
  • anonymous
can u continue from here?
ParthKohli
  • ParthKohli
\[f(10) = 10^3 - 3(10)^2 +9(10) -13 = 1000 - 300 + 90 - 13 \\ = 777 \]
anonymous
  • anonymous
thats correct
anonymous
  • anonymous
\[\huge \color{green}{THANKS}\]
anonymous
  • anonymous
welcome

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