Thread: Functional Equation

Greetings,

I am not quite sure what classification this problem should actually have, but my teacher, who was unable to help me with it, said it was a functional equation.

The aim is to find the values of a, b, c and d or represent the function on the first line like the one on the second line.

Originally Posted by Logic

Greetings,

I am not quite sure what classification this problem should actually have, but my teacher, who was unable to help me with it, said it was a functional equation.

The aim is to find the values of a, b, c and d or represent the function on the first line like the one on the second line.

This is just like finding the Taylor series of f center at x=1.

$\displaystyle f(x)=a(x-1)^3+b(x-1)^2+c(x-1)+d$


If we take a few derivatives we will see that

$\displaystyle f'(x)=3a(x-1)^2+2b(x-1)+c$

$\displaystyle f''(x)=6a(x-1)+2b$

$\displaystyle f'''(x)=6a$

Now compare this with the other functions derivatives

$\displaystyle f(x)=x^3+x^2-x+1$
$\displaystyle f'(x)=3x^2+2x-1$
$\displaystyle f''(x)=6x+2$
$\displaystyle f'''(x)=6$

Now compare the derivatives

$\displaystyle f'''(1)=6 =6a \implies a=1$

$\displaystyle f''(1)=6+2=2b \implies b=4$

$\displaystyle f'(1)=4=c \implies c=4$

$\displaystyle f(1)=2=d$

So we get

$\displaystyle f(x)=(x-1)^3+4(x-1)^2+4(x-1)+2$