First slide

Algebra of real valued functions

Question

$If a function f : R \to R satisfies$ $(x - y) f (x + y) - (x + y) f (x - y) = 2 (x^{2} y - y^{3}) \forall x, y \in R and f (1) = 2, then$

Difficult

A

f(x) must be polynomial function
B

f(3) = 12
C

f(0) = 0
D

f(x) may not be differentiable

Solution

(A,B,C)
$\begin{array}{l} (x - y) f (x + y) - (x + y) f (x - y) = 2 y (x - y) (x + y) \\ Let x - y = u; x + y = v \\ u f (v) - v f (u) = u v (v - u) \\ \frac{f (v)}{v} - \frac{f (u)}{u} = v - u \end{array}$
$\begin{array}{l} (\frac{f (v)}{v} - v) = (\frac{f (u)}{u} - u) = constant \\ Let \frac{f (x)}{x} - x = λ \Rightarrow f (x) = (λ x + x^{2}) \\ f (1) = 2 \\ λ + 1 = 2 \Rightarrow λ = 1 f (x) = x^{2} + x \end{array}$

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