For x [–14,13] the function f is defined by
f(x)=(x^3)(x+6)^4
On which two intervals is the function increasing (enter intervals in ascending order)?
I got increasing intervals between [-14, -6], but can't find the other increasing interval. I believe it is [-3, 13] but I get the answer is wrong. Thanks again.
The places where the function changes from increasing to decreasing are where f'(x) = 0
f'(x) = 3x^2(x+6)^4 + 4(x+6)^3*x^3
= (x+6)^3 [7x^3 +18x^2]
I can see that becoming zero at x=0, x=-6, and x = -18/7, but not at x=-3
recheck your numbers and mine
To determine the intervals on which the function is increasing, we need to analyze the sign of the derivative of the function.
First, let's find the derivative of the function f(x). We will use the power rule for differentiation:
f'(x) = (3x^2)(x+6)^4 + (x^3)(4)(x+6)^3
Now, we need to determine where the derivative is positive (greater than 0) to identify the increasing intervals.
To simplify the analysis, we will look at the sign of each factor separately.
1. (3x^2)(x+6)^4
The factor (3x^2) is positive for all x, as (3x^2) is positive when x is positive or negative.
The factor (x+6)^4 is also positive for all x, as it is a power of x+6, and any non-zero value raised to an even power is positive.
Therefore, the product of these two factors is positive for all x.
2. (x^3)(4)(x+6)^3
The factor (x^3) is positive for positive x and negative for negative x.
The factor (4) is positive.
The factor (x+6)^3 is positive for x > -6 and negative for x < -6 (as it is a power of (x+6)).
Now, let's analyze the different intervals:
1. For x ≤ -6 (from -14 to -6)
- All factors are positive, so the function f(x) is increasing on this interval.
2. For -6 < x ≤ -3 (from -6 to -3)
- The factor (3x^2)(x+6)^4 is positive, so the function f(x) is increasing on this interval.
- The factor (x^3)(4)(x+6)^3 is negative, so it does not contribute to the function being increasing.
3. For x > -3 (from -3 to 13)
- The factor (3x^2)(x+6)^4 is positive, so the function f(x) is increasing on this interval.
- The factor (x^3)(4)(x+6)^3 is positive when x > -6, so it also contributes to the function being increasing.
From the analysis above, we can conclude that the function f(x) is increasing on two intervals: [-14, -6] and (-3, 13]. The intervals are expressed in ascending order.
Note: The interval -6 < x ≤ -3 is not included because the function is not strictly increasing on that interval. It is considered constant since the factor (x^3)(4)(x+6)^3 is negative.