Answer:
The function
is decreasing on the interval
and it is increasing on the interval ![(-\infty, -1)\cup (7, \infty)](https://tex.z-dn.net/?f=%28-%5Cinfty%2C%20-1%29%5Ccup%20%287%2C%20%5Cinfty%29)
Step-by-step explanation:
To determine the intervals of increase and decrease of the function
, perform the following steps:
1. Differentiate the function
![\frac{d}{dx}\left(6x^3-54x^2-126x-8\right)=\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\\frac{d}{dx}\left(6x^3\right)-\frac{d}{dx}\left(54x^2\right)-\frac{d}{dx}\left(126x\right)-\frac{d}{dx}\left(8\right)\\\\f'(x)=18x^2-108x-126](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%286x%5E3-54x%5E2-126x-8%5Cright%29%3D%5C%5C%5C%5C%5Cmathrm%7BApply%5C%3Athe%5C%3ASum%2FDifference%5C%3ARule%7D%3A%5Cquad%20%5Cleft%28f%5Cpm%20g%5Cright%29%27%3Df%5C%3A%27%5Cpm%20g%27%5C%5C%5C%5C%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%286x%5E3%5Cright%29-%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%2854x%5E2%5Cright%29-%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%28126x%5Cright%29-%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%288%5Cright%29%5C%5C%5C%5Cf%27%28x%29%3D18x%5E2-108x-126)
2. Obtain the roots of the derivative, f'(x) = 0
![\mathrm{Factor\:out\:common\:term\:}18:\quad 18\left(x^2-6x-7\right)\\\\\mathrm{Factor}\:x^2-6x-7:\quad \left(x+1\right)\left(x-7\right)\\\\18x^2-108x-126:\quad 18\left(x+1\right)\left(x-7\right)\\\\18x^2-108x-126=0\quad :\quad x=-1,\:x=7](https://tex.z-dn.net/?f=%5Cmathrm%7BFactor%5C%3Aout%5C%3Acommon%5C%3Aterm%5C%3A%7D18%3A%5Cquad%2018%5Cleft%28x%5E2-6x-7%5Cright%29%5C%5C%5C%5C%5Cmathrm%7BFactor%7D%5C%3Ax%5E2-6x-7%3A%5Cquad%20%5Cleft%28x%2B1%5Cright%29%5Cleft%28x-7%5Cright%29%5C%5C%5C%5C18x%5E2-108x-126%3A%5Cquad%2018%5Cleft%28x%2B1%5Cright%29%5Cleft%28x-7%5Cright%29%5C%5C%5C%5C18x%5E2-108x-126%3D0%5Cquad%20%3A%5Cquad%20x%3D-1%2C%5C%3Ax%3D7)
3. Form open intervals with the roots of the derivative and take a value from every interval and find the sign they have in the derivative.
If f'(x) > 0, f(x) is increasing.
If f'(x) < 0, f(x) is decreasing.
On the interval
, take x = -2,
f'(x) > 0 therefore f(x) is increasing
On the interval
, take x = 0,
f'(x) < 0 therefore f(x) is decreasing
On the interval
, take x = 10,
f'(x) > 0 therefore f(x) is increasing
The function
is decreasing on the interval
and it is increasing on the interval ![(-\infty, -1)\cup (7, \infty)](https://tex.z-dn.net/?f=%28-%5Cinfty%2C%20-1%29%5Ccup%20%287%2C%20%5Cinfty%29)