Question

f(x) is diff fⁿ

$f(\frac{x+2y}{3}) = \frac{f(x)+2f(y)}{3}$, $f(0)=3$, $f'(0)=2$

Find f(x)

Answer 1

f(x) = 2x+3

Answer 2

Differentiate the functional equation with respect to y: $f'(\frac{x+2y}{3}) \cdot \frac{2}{3} = \frac{2f'(y)}{3}$ $f'(\frac{x+2y}{3}) = f'(y)$ Set y=0: $f'(\frac{x}{3}) = f'(0) = 2$. This means $f'(z) = 2$ for all z. Integrate $f'(x)=2$ to get $f(x) = 2x+C$. Using $f(0)=3$: $f(0) = 2(0)+C = 3 \implies C=3$. Thus, $f(x) = 2x+3$.