Practicing Success
Let f : R → R be a function satisfying $f(x+y)=f(x)+2 y^2+k x y$ for all $x, y \in R$. If f(1) = 2 and f(2) = 8, then f(x) is equal to |
$2 x^2$ $6 x-4$ $x^2+3 x-2$ $-x^2+9 x-6$ |
$2 x^2$ |
$f(x+y)=f(x)+2 y^2+k x y$ for all $x, y \in R$ $\Rightarrow \frac{f(x+y)-f(x)}{y}=2 y+k x$ for all $x \in R$ $\Rightarrow \lim\limits_{y \rightarrow 0} \frac{f(x+y)-f(x)}{y}=\lim\limits_{y \rightarrow 0}(2 y+k x)$ $\Rightarrow f'(x)=k x$ for all $x \in R$ $\Rightarrow f(x)=\frac{k x^2}{2}+C$ for all $x \in R$ But, f(1) = 2 and f(2) = 8. Therefore, 2 = $\frac{k}{2}$ + C and 8 = 2k + C ⇒ k = 4 and C = 0 Hence, f(x) = 2x2 for all $x \in R$. |