Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$ . A)Increasing on x 2,decreasing on -3

Question

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$ . A)Increasing on x < -3 and x > 2,decreasing on -3 < x < 2 B)Increasing on -3 < x < 2,decreasing on x < -3 and x > 2 C)Increasing on x < -2,decreasing on x > 3 D)Increasing on x $\leq$ -2 and x $\geq$ 3 ,decreasing on - 2 $\leq$ x $\leq$ 3

Quizplus · Accepted Answer

To determine the intervals of increase and decrease for the function $f(x) = 6x^3 + 9x^2 - 108x - 3$, we first find its derivative, $f'(x) = 18x^2 + 18x - 108$. Factoring out 18 gives $f'(x) = 18(x^2 + x - 6) = 18(x + 3)(x - 2)$. Setting $f'(x) = 0$ gives the critical points $x = -3$ and $x = 2$. Testing intervals around these points in the derivative shows that $f'(x) > 0$ (increasing) for $x < -3$ and $x > 2$, and $f'(x) < 0$ (decreasing) for $-3 < x < 2$.

Find the Intervals of Increase and Decrease For f(x)=6x3+9x2−108x−3f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3f(x)=6x3+9x2−108x−3

Find the Intervals of Increase and Decrease For $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$