For x including [–15, 13] the function f is defined by f(x)=x^7(x+4)^6?
For x including [–15, 13] the function f is defined by
f(x)=x^7(x+4)^6
On which two intervals is the function increasing (enter intervals in ascending order)?
( i was able to get the first interval but had problems with the first number on the second. I thought it was zero but its wrong)
1.( -15 to -4)
and
2. (???? to 13)
To determine the intervals on which the function is increasing, we need to find where the derivative of the function is positive. Let's calculate the derivative of the function f(x) = x^7(x+4)^6 first.
Step 1: Calculate the derivative of the function f(x).
Using the power rule and the product rule of differentiation, we can find the derivative of f(x) as follows:
f'(x) = 7x^6(x+4)^6 + x^7 * 6(x+4)^5
Simplifying further:
f'(x) = 7x^6(x+4)^6 + 6x^7(x+4)^5
Step 2: Set the derivative equal to zero and solve for x.
To find where the derivative is positive, we need to solve the equation f'(x) = 0. However, in this case, we don't need to solve the equation since the function is in the form (x+4)^n, so it will never equal zero within the given interval [-15, 13].
Step 3: Determine the intervals based on the sign of the derivative.
Since the equation f'(x) = 0 has no solutions in the given interval, we need to check the sign of the derivative in specific points to determine the intervals.
Let's consider three points: x = -15, x = -4, and x = 13.
At x = -15:
Evaluate f'(-15) = 7(-15)^6(-15+4)^6 + 6(-15)^7(-15+4)^5. The value of this expression doesn't matter to us. What matters is whether it is positive or negative. Let's say this value is negative.
At x = -4:
Evaluate f'(-4) = 7(-4)^6(-4+4)^6 + 6(-4)^7(-4+4)^5. Since (-4)^6 is positive and (-4+4) is zero, the value of f'(-4) is also zero. This point is a potential turning point for the function.
At x = 13:
Evaluate f'(13) = 7(13)^6(13+4)^6 + 6(13)^7(13+4)^5. The value of this expression doesn't matter to us. What matters is whether it is positive or negative. Let's say this value is positive.
Step 4: Analyze the intervals based on the sign.
- From x = -15 to x = -4, the function is increasing, as the derivative is negative or zero.
- From x = -4 to x = 13, we cannot determine if the function is increasing or decreasing since we have a potential turning point at x = -4.
- From x = 13 to the end of the given interval, the function is increasing, as the derivative is positive.
Therefore, the intervals on which the function f(x) = x^7(x+4)^6 is increasing are:
1. [-15, -4]
2. [-4, 13]
(Note that we don't have a specific endpoint for the second interval, as the function might change at x = -4, but as given, we can only determine that the function is increasing from -4 onwards.)