What is the length of the largest interval on which the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ is decreasing?
A) $3 \sqrt { 2 }$
B)1
C)4
D) $2 \sqrt { 2 }$
E)3
F)
$2 \sqrt { 3 }$
G) $\sqrt { 3 }$
H)2

Question

What is the length of the largest interval on which the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ is decreasing?
A) $3 \sqrt { 2 }$ 
B)1
C)4
D) $2 \sqrt { 2 }$
E)3
F) 
$2 \sqrt { 3 }$ 
G) $\sqrt { 3 }$  
H)2

Quizplus · Accepted Answer

To determine the interval where the function $f(x) = x^3 - 3x^2 - 9x$ is decreasing, we first find its derivative, $f'(x) = 3x^2 - 6x - 9$. Setting $f'(x) = 0$ to find critical points, we solve $3x^2 - 6x - 9 = 0$, which simplifies to $x^2 - 2x - 3 = 0$. Factoring or using the quadratic formula, we find the roots $x = 1 - \sqrt{4}$ and $x = 1 + \sqrt{4}$, which are $x = -1$ and $x = 3$, respectively. The function is decreasing where its derivative is negative, which, by testing intervals, is between $x = -1$ and $x = 3$. Therefore, the length of the largest interval on which the function is decreasing is $3 - (-1) = 4$.

What Is the Length of the Largest Interval on Which f(x)=x3−3x2−9xf ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 xf(x)=x3−3x2−9x

What Is the Length of the Largest Interval on Which $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$