로그함수의 최댓값

$f ( x ) = \log _{9} ( 5 - x ) + \log _{3} ( x + 4 )$ $\displaystyle = \frac{1}{2} \log _{3} ( 5 - x ) + \log _{3} ( x + 4 )$ $\displaystyle = \frac{1}{2} \left\{ \log _{3} ( 5 - x ) + 2 \log _{3} ( x + 4 ) \right\}$ $\displaystyle = \frac{1}{2} \log _{3} ( 5 - x ) ( x + 4 ) ^{2}$ 진수 조건에서 $5 - x > 0 , x + 4 > 0$ $\rightarrow - 4 < x < 5$ $g ( x ) = ( 5 - x ) ( x + 4 ) ^{2}$이라 놓으면 $g' ( x ) = - ( x + 4 ) ^{2} + ( 5 - x ) \cdot 2 ( x + 4 )$$= - 3 ( x - 2 ) ( x - 4 )$ $- 4 < x < 5$에서 $g' ( x ) = 0$인 $x$의 값은 $x = 2$ 증감을 조사하면 $g ( x )$는 $x = 2$에서 극대이면서 최대이고, 최댓값은 $g ( 2 ) = 3 \cdot 6 ^{2}$$= 108$ $f ( x )$의 최댓값은 $\displaystyle \frac{1}{2} \log _{3} 108$$\displaystyle = \frac{1}{2} \log _{3} ( 3 ^{3} \cdot 2 ^{2} )$$\displaystyle = \frac{1}{2} ( 3 + 2 \log _{3} 2 )$$\displaystyle = \frac{3}{2} + \log _{3} 2$