Rewrite the expression as a sum, difference, or multiple of logarithms.
-Find where $f ^ { \prime } ( x ) = 0$ if $f ^ { \prime } ( x ) = \left( x ^ { 2 } - 9 \right) \left[ 2 ( x - 8 ) ^ { 2 } \right] + ( x - 8 ) ^ { 3 } ( 5 x )$
A) $x = - 8 , x = \frac { 40 + \sqrt { 526 } } { 7 } , x = \frac { 40 - \sqrt { 526 } } { 7 }$
B) $x = 8 , x = \frac { 20 + \sqrt { 526 } } { 7 } , x = \frac { 20 - \sqrt { 526 } } { 7 }$
C)$ \mathrm { x } = 8 , \mathrm { x } = \frac { 20 + \sqrt { 526 } } { 14 } , \mathrm { x } = \frac { 20 - \sqrt { 526 } } { 14 }$
D) $x = 8 , x = \frac { 40 + 2 \sqrt { 1052 } } { 7 } , x = \frac { 40 - 2 \sqrt { 1052 } } { 7 }$

Question

Quizplus · Accepted Answer

To solve for $f'(x) = 0$, we first simplify the given expression. Factoring out common terms and solving the quadratic equation will lead to the roots provided in option B. The key is recognizing that $x = 8$ is a root due to the $(x - 8)^2$ and $(x - 8)^3$ terms, and solving the quadratic part of the equation gives the other two roots.

Rewrite the Expression as a Sum, Difference, or Multiple of Logarithms