함수방정식과 미분

$f ( x + y ) = f ( x ) + f ( y ) + 2 xy - 1$에서 $x = 0 , y = 0$을 대입하면 $f ( 0 ) = f ( 0 ) + f ( 0 ) - 1$ ∴ $f ( 0 ) = 1$ $\displaystyle f' ( 0 ) = \lim\limits _{h\rightarrow 0} \frac{f ( h ) - f ( 0 )}{h} = \lim\limits _{h\rightarrow 0} \frac{f ( h ) - 1}{h}$ 이므로 $\displaystyle f' ( x ) = \lim\limits _{h\rightarrow 0} \frac{f ( x + h ) - f ( x )}{h}$ $\displaystyle = \lim\limits _{h\rightarrow 0} \frac{f ( x ) + f ( h ) + 2 xh - 1 - f ( x )}{h}$ $\displaystyle = 2 x + \lim\limits _{h\rightarrow 0} \frac{f ( h ) - 1}{h}$ $= 2 x + f' ( 0 )$ $\cdots \text{㉠}$ $\displaystyle \lim\limits _{x\rightarrow 1} \frac{f ( x ) - f' ( x )}{x ^{2} - 1} = 14$에서 $x \rightarrow 1$일 때, (분모)$\rightarrow 0$이므로 (분자)$\rightarrow 0$이어야 한다. ∴ $f ( 1 ) = f' ( 1 )$ ㉠에서 $f' ( 1 ) = 2 + f' ( 0 )$이므로 $f' ( 0 ) = f' ( 1 ) - 2 = f ( 1 ) - 2$ ∴ $\displaystyle \lim\limits _{x\rightarrow 1} \frac{f ( x ) - f' ( x )}{x ^{2} - 1}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{f ( x ) - 2 x - f' ( 0 )}{x ^{2} - 1}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{f ( x ) - 2 x - f ( 1 ) + 2}{x ^{2} - 1}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{f ( x ) - f ( 1 )}{x ^{2} - 1} - \lim\limits _{x\rightarrow 1} \frac{2 ( x - 1 )}{x ^{2} - 1}$ $\displaystyle = \frac{1}{2} f' ( 1 ) - 1$ $= 14$ ∴ $f' ( 1 ) = 30$ ∴ $f' ( 0 ) = f' ( 1 ) - 2 = 28$ [다른풀이] $f ( x + y ) = f ( x ) + f ( y ) + 2 xy - 1$에서 $x = 0 , y = 0$을 대입하면 $f ( 0 ) = f ( 0 ) + f ( 0 ) - 1$ ∴ $f ( 0 ) = 1$ $f' ( 0 ) = k$라 하면 $\displaystyle k = \lim\limits _{h\rightarrow 0} \frac{f ( h ) - f ( 0 )}{h} = \lim\limits _{h\rightarrow 0} \frac{f ( h ) - 1}{h}$ $\displaystyle f' ( x ) = \lim\limits _{h\rightarrow 0} \frac{f ( x + h ) - f ( x )}{h}$ $\displaystyle = \lim\limits _{h\rightarrow 0} \frac{f ( x ) + f ( h ) + 2 xh - 1 - f ( x )}{h}$ $\displaystyle = 2 x + \lim\limits _{h\rightarrow 0} \frac{f ( h ) - 1}{h}$ $= 2 x + k$ ∴ $\displaystyle f ( x ) = \int ( 2 x + k ) dx = x ^{2} + kx + C$ $( C$는 상수) $f ( 0 ) = 1$이므로 $C = 1$ ∴ $f ( x ) = x ^{2} + kx + 1$ 따라서, $f' ( x ) = 2 x + k$이므로 $\displaystyle \lim\limits _{x\rightarrow 1} \frac{f ( x ) - f' ( x )}{x ^{2} - 1} = \frac{\lim\limits _{x\rightarrow 1{x ^{2} + kx + 1 - 2 x - k}}}{x ^{2} - 1}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{( x - 1 ) ^{2} + k ( x - 1 )}{( x - 1 ) ( x + 1 )}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{x - 1 + k}{x + 1}$ $\displaystyle = \frac{k}{2}$ $= 14$ ∴ $k = 28$