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Jonask
 2 years ago
functional equations
Jonask
 2 years ago
functional equations

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Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1\[\color{brown}{f(x)+(x+1)^3=2f(x+1)}\]

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0Find \(\color{#C00}{f(10)}\)

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1lol thanks for posting the full question @ParthKohli

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1wat to do next i 3 variables 2 equations??

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0\[f(0) = 2f(1)  1 \\ f(1) = 2f(2)  8\]So\[f(0) = 2(2f(2)  8) = 4f(2)  16\]

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1\[f(0)=4(f(2)8)1=4f(2)161=4f(2)17\]

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1is f always of the from\[ax^2+bx+c\]

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0No, it's a polynomial. It can be in that form, but we're not sure.

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1so far we see \[f(10)+11^3=2f(11)\] \[f(9)+10^3=2f(10)\]  \[f(10)=f(9)2f(11)+11^310^3\]

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0I think that it must be in the form \(ax^3 + bx^2 + cx + d\)

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1we need a relation between f(9) and f(11)

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0\[f(9) = 2f(10)  1000\]\[f(10) = 2f(11)  1331\]\[\iff f(9) = 2(f(11)  1331 )  1000\]

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0\[f(9) = 2f(11)  2662  1000\]

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0That's the relation.

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1isnt this the solution then

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1since \[f(9)2f(11)=3662\] \[f(10)=3662+11^310^3\]

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0But it's an integer between 0 and 999.

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1okay so we found a relationship between numbers 2 units away from each other x1 and x+1 \[\color{blue}{f(x1)+x^3=2f(x).......... f(x)+(1+x)^3=2f(1+x)}\] \[f(x1)+x^3=4f(x+1)+2(x+1)^3\]

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1\[f(9)4f(11)=2(11)^310^3\]

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.3Let f(x)=a+bx+cx^2+dx^3 NOTE: f(x) must be a third degree polynomial

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.3Now, f(x)+(x+1)^3 =2f(x+1)

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1\[\huge f(10)=f(9)2f(11)+11^310^3=???\]

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.3use it to find a,b,c and d

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.3u will get a=13 b=9 c=3 d=1

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0Equating coefficients

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.3now, once u know f(x) u can find f(10)

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1help me i dont get where you get the coeeffients from

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0@Jonask a technique.\[ax + by = 2x + 3y\]means\[a=2,b=3\]

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.3f(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

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.3can u continue from here?

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0\[f(10) = 10^3  3(10)^2 +9(10) 13 = 1000  300 + 90  13 \\ = 777 \]

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.1\[\huge \color{green}{THANKS}\]
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