6)
The function
f(x)=4x3−18x2−480x−2
is decreasing on the interval ?
Enter your answer using the interval notation for open intervals.
It is increasing on the interval(s) ?
The function has a local maximum at ?
To determine when a function is decreasing or increasing, we need to analyze its derivative. Let's find the derivative of the given function first:
f(x) = 4x^3 - 18x^2 - 480x - 2
To find the derivative, we differentiate each term individually using the power rule:
f'(x) = (4 * 3)x^(3 - 1) - (18 * 2)x^(2 - 1) - (480 * 1)x^(1 - 1)
= 12x^2 - 36x - 480
Now that we have the derivative, we can determine when the function is decreasing or increasing.
When the derivative is negative (f'(x) < 0), the function is decreasing. Similarly, when the derivative is positive (f'(x) > 0), the function is increasing.
For this question, we need to find the intervals where we have either a decrease or an increase.
To find the intervals of increasing and decreasing, we need to find the critical points of the function. Critical points occur where the derivative equals zero or is undefined.
Setting f'(x) to zero and solving for x gives us:
12x^2 - 36x - 480 = 0
Factoring out common terms:
12(x^2 - 3x - 40) = 0
Now, using the zero-product property, we solve for x:
x^2 - 3x - 40 = 0
(x - 8)(x + 5) = 0
Setting each factor to zero:
x - 8 = 0 or x + 5 = 0
So, x = 8 or x = -5 are the critical points of the function.
To determine the intervals of increase and decrease, we can use a sign chart or test points within each interval.
First, let's test a point less than -5, such as x = -6:
f'(-6) = 12(-6)^2 - 36(-6) - 480
= 432 + 216 - 480
= 168
Since f'(-6) > 0, the function is increasing to the left of -5.
Now, let's test a point between -5 and 8, such as x = 0:
f'(0) = 12(0)^2 - 36(0) - 480
= 0
Since f'(0) = 0, this could indicate a local maximum or a saddle point. We need to further analyze this point.
Finally, let's test a point greater than 8, such as x = 9:
f'(9) = 12(9)^2 - 36(9) - 480
= 972 - 324 - 480
= 168
Since f'(9) > 0, the function is increasing to the right of 8.
Based on the analysis, we have the following information:
- The function is decreasing on the interval (-∞, -5).
- The function is increasing on the interval (-5, 8) and (8, +∞).
- The function has a local maximum at x = 0, as f'(0) = 0.
Therefore, the function is decreasing on the interval (-∞, -5), and it is increasing on the interval (-5, 8) and (8, +∞). Additionally, the function has a local maximum at x = 0.