Calc HW help! For x E[-14,11] the function is defined by f(x)=x^3(x+7)^6.

Question

On which two intervals is the function increasing?

Answer 1

The Function is 

Derivative of the function 

For x = 0,x = -7, x= -7 / 3  the value of .

Hence they are called as stationary points of f(x).

Case 1)    -14<x < -7

The first term 3x^2 is positive .

The second term (x+7)^5 is  negative.

The third term (3x +7) is negative.

So, in this interval the function is positive that is in this interval the function is increasing.

Case 2)  -7 < x < -7/3

The first term (3x^2) is positive.

The second term (x +7)^ 5 is  positive.

The third term (3x+7)  negative.

So,in this interval the function is negative that is in this interval the function is decreasing.

Case 3 : -7/3<x<11

The first term (3x^2) is positive.

The second term (x+7)^5 is positive

The third term (3 x +7)  is positive.

Hence the function is positive in this interval that is in this interval the function is increasing.

So,for the  intervals -14 <x< 7 and -7/3<x<11,the function is increasing.

Hence the two intervals for which the function is increasing are [-14 ,7] and[-7/3,11] .