함수방정식과 극값

$f ( x - y ) = f ( x ) - f ( y ) + xy ( x - y )$에 $x = 0$, $y = 0$을 대입하면 $f ( 0 ) = f ( 0 ) - f ( 0 )$ ∴ $f ( 0 ) = 0$ ㉠ $f' ( 0 ) = 8$이므로 $\displaystyle f' ( 0 ) = \lim\limits _{h \rightarrow 0} \frac{f ( h ) - f ( 0 )}{h} = \lim\limits _{h \rightarrow 0} \frac{f ( h )}{h} ( \because \text{㉠} ) = 8$ ㉡ ∴ $\displaystyle f' ( x ) = \lim\limits _{h \rightarrow 0} \frac{f ( x + h ) - f ( x )}{h}$ $\displaystyle = \lim\limits _{h \rightarrow 0} \frac{f ( x ) - f ( - h ) + x \cdot ( - h ) ( x + h ) - f ( x )}{h}$ (∵ (가)) $\displaystyle = \lim\limits _{h \rightarrow 0} \left\{ \frac{f ( - h )}{- h} - x ^{2} - xh \right\}$ $= 8 - x ^{2} ( \because \text{㉡} )$ $\displaystyle = ( 2 \sqrt{2} + x ) ( 2 \sqrt{2} - x )$ 함수 $f ( x )$가 $x = a$에서 극대값을 갖고 $x = b$에서 극소값을 가지므로 $\displaystyle a = 2 \sqrt{2}$, $\displaystyle b = - 2 \sqrt{2}$ ∴ $a ^{2} + b ^{2} = 8 + 8 = 16$