Question

Given $f(x)=-\frac{x^3}{3}+x^2 \sin 1.5 a-x \sin a \cdot \sin 2 a-5 \arcsin \left(a^2-8 a+17\right)$ then

• A. $f(x)$ is not defined at $x=\sin 8$
• B. $f ^{\prime}(\sin 8)>0$
• C. $f ^{\prime}( x )$ is not defined at $x =\sin 8$
• D. $ f ^{\prime}(\sin 8)<0$

Answer 1

Answer: (D)

$ f(x)=-\frac{x^3}{3}+x^2 \sin 6-x \sin 4 \cdot \sin 8-5 \sin ^{-1} \left((a-4)^2+1\right) $
$f ^{\prime}( x )=- x ^2+2 x \sin 6-\sin 4 \sin 8 ( a =4) $
$f^{\prime}(\sin 8)=-\sin ^2 8+2 \sin 6 \sin 8-\sin 4 \sin 8$
$=\sin 8[-\sin 8+2 \sin 6-\sin 4]$
$=-\sin 8[\sin 8+\sin 4-2 \sin 6]=-\sin 8[2 \sin 6 \cos 2-2 \sin 6]$
$=2 \sin 8 \sin 6[1-\cos 2]$