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 $\le$ -2 and x $\le$3 , decreasing on - 2 $\le$ x $\le$ 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, we get $f'(x) = 18(x^2 + x - 6)$. Factoring the quadratic, we get $f'(x) = 18(x + 3)(x - 2)$. Setting $f'(x) = 0$ gives us critical points at $x = -3$ and $x = 2$. Testing intervals around these points in the derivative tells us about the function's behavior: it 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−3f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3f(x)=6x3+9x2−108x−3

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