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

Question

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

Quizplus · Accepted Answer

To determine the intervals of increase and decrease for the function $f(x) = 6x^3 + 9x^2 - 108x - 2$, we first find its derivative, $f'(x) = 18x^2 + 18x - 108$. Factoring this 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 values in the intervals $(-\infty, -3)$, $(-3, 2)$, and $(2, \infty)$ in the derivative $f'(x)$ will show that the function is increasing when $x < -3$ and $x > 2$, and decreasing when $-3 < x < 2$.

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

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