# Functional Equation

• Jan 18th 2010, 10:33 AM
Logic
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.
• Jan 18th 2010, 06:26 PM
TheEmptySet
Quote:

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\$