함수방정식과 미분

$y$대신 0대입$\rightarrow f ( x ) = f ( x ) + f ( 0 ) - 3 \rightarrow \therefore f ( 0 ) = 3$ i) 도함수의 정의를 활용하면 ii) 편미분을 활용하면$\rightarrow$$y$만 변수 취급 $\displaystyle f' ( x ) = \lim\limits _{h \rightarrow 0} {\frac{f ( x + h ) - f ( x )}{h}}$ $f' ( x + y ) = f' ( y ) + x ^{2} + 2 xy$ $\displaystyle = \lim\limits _{h \rightarrow 0} {\frac{f ( x ) + f ( h ) + xh ( x + h ) - 3 - f ( x )}{h}}$ $\downarrow$$y$대신 0대입 $\displaystyle = \lim\limits _{h \rightarrow 0} {\left. \left( \frac{f ( h ) - 3}{h} \right. + x ( x + h ) \right)}$ $\therefore f' ( x ) = x ^{2} + f' ( 0 )$ $\displaystyle = f' ( 0 ) + x ^{2} \left( {\because f ( 0 ) = 3 , \lim\limits _{h \rightarrow 0} {x ( x + h ) = x ^{2}} )} \right.$ 문제에서 $f' ( 1 ) = 2$이므로 대입하면 $f' ( 1 ) = 1 + f' ( 0 ) = 2 \rightarrow \therefore f' ( 0 ) = 1$ $\therefore f' ( x ) = x ^{2} + 1$ 이 식을 적분하면 $\displaystyle f ( x ) = \frac{1}{3} x ^{3} + x + 3 ( \because f ( 0 ) = 3 )$ $\rightarrow$ $\displaystyle \therefore f ( 3 ) = \frac{1}{3} \cdot ( 3 ) ^{3} + 3 + 3 = 15$