함수방정식과 미분

$y = 0$을 대입하면 $f ( x ) = f ( x ) f ( 0 ) + 4 f ( x ) + 4 f ( 0 ) + 12$ $\left\{ f ( x ) + 4 \right\} \left\{ f ( 0 ) + 3 \right\} = 0$ $f ( 0 ) = - 3$ $\displaystyle f' ( x ) = \lim\limits _{h \rightarrow 0} {\frac{f ( x + h ) - f ( x )}{h}}$ $\displaystyle = \lim\limits _{h \rightarrow 0} {\frac{f ( x ) f ( h ) + 4 f ( x ) + 4 f ( h ) + 12 - f ( x )}{h}}$ $\displaystyle = \lim\limits _{h \rightarrow 0} {\frac{f ( x ) f ( h ) + 3 f ( x ) + 4 f ( h ) + 12}{h}}$ $\displaystyle = \lim\limits _{h \rightarrow 0} {\frac{\left\{ f ( x ) + 4 \right\} \left\{ f ( h ) + 3 \right\}}{h}}$ $\displaystyle = {\left\{ f ( x ) + 4 \right\} \lim\limits _{h \rightarrow 0} {\frac{f ( h ) + 3}{h}}}$ $\displaystyle = \left\{ f ( x ) + 4 \right\} \lim\limits _{h \rightarrow 0} {\frac{f ( h ) - f ( 0 )}{h}}$ $= \left\{ f ( x ) + 4 \right\} f' ( 0 )$ $= 2 \left\{ f ( x ) + 4 \right\}$ $\therefore$$f' ( \ln 2 ) = 2 \left\{ f ( \ln 2 ) + 4 \right\} = 8$ [다른 풀이] (가)에서 양변에 $4$를 더하여 인수분해하면 $f ( x + y ) + 4 = \left\{ f ( x ) + 4 \right\} \left\{ f ( y ) + 4 \right\}$ $f ( x ) + 4 = g ( x )$라 하면 $g ( x + y ) = g ( x ) g ( y )$ …㉠ ㉠에서 $y = 0$이라 하면 $g ( x ) = g ( x ) g ( 0 )$ $g ( x ) \left\{ 1 - g ( 0 ) \right\} = 0$ $g ( 0 ) = 1$ $f' ( x ) = g' ( x )$ $\displaystyle = \lim\limits _{h \rightarrow 0} {\frac{g ( x + h ) - g ( x )}{h}}$ $\displaystyle = \lim\limits _{h \rightarrow 0} {\frac{g ( x ) g ( h ) - g ( x )}{h}}$ $\displaystyle = g ( x ) \lim\limits _{h \rightarrow 0} {\frac{g ( h ) - 1}{h}}$ $\displaystyle = g ( x ) \lim\limits _{h \rightarrow 0} {\frac{g ( h ) - g ( 0 )}{h}}$ $= g ( x ) g' ( 0 )$ $( \because g' ( 0 ) = f' ( 0 ) = 2 )$ $= 2 g ( x ) = 2 \left\{ f ( x ) + 4 \right\}$ 따라서 $f' ( \ln 2 ) = 2 \left\{ f ( \ln 2 ) + 4 \right\}$$= 2 \times 4$$= 8$