Question
For the following function, find the intervals on which the function is increasing or decreasing.
List any interval(s) on which the function is increasing. Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The function is increasing on the interval(s)
(Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers
as needed.)
B. The function is never increasing.
List any interval(s) on which the function is decreasing. Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The function is decreasing on the interval(s)
(Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers
as needed.)
B. The function is never decreasing.
List any interval(s) on which the function is increasing. Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The function is increasing on the interval(s)
(Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers
as needed.)
B. The function is never increasing.
List any interval(s) on which the function is decreasing. Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The function is decreasing on the interval(s)
(Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers
as needed.)
B. The function is never decreasing.
Ask by Bartlett Little. in the United States
Mar 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The function is increasing on the intervals
Solution
- Compute the derivative of the function:
- Solve for critical points by setting
:
Divide the entire equation by 6:
Use the quadratic formula:
Thus, the critical points are:
- Analyze the sign of
on the intervals determined by the critical points:
-
For
: Choose Hence,is increasing on this interval. -
For
: Choose Hence,is decreasing on this interval. -
For
: Choose Hence,is increasing on this interval.
- Final answer in interval notation:
- The function is increasing on the interval(s):
- The function is decreasing on the interval(s):
Answered by UpStudy AI and reviewed by a Professional Tutor
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Explain 
Simplify this solution The Deep Dive
To determine where the function
First, the derivative
Using the quadratic formula,
After applying the formula, we find the critical points, and then test the intervals around these points to find where the derivative is positive (indicating increasing behavior) or negative (indicating decreasing behavior).
By evaluating the derivative around the critical points, we discover that:
- The function is increasing on the interval(s):
. - The function is decreasing on the interval(s):
.
So here’s how you might fill in your answer:
A. The function is increasing on the interval(s)
A. The function is decreasing on the interval(s)
Now, let’s sprinkle in some extra knowledge!
The polynomial function you peeked at has a lot of character! It’s shaped like a roller coaster, creasing and dipping around its critical points. So, every time you analyze it mathematically, imagine it feels those ups and downs—like an adventure ride in the world of functions!
If you’re intrigued by more complex functions, you could dive into topics like calculus of higher-order derivatives or even explore how these principles apply to real-world scenarios like optimizing profit in business or modeling population dynamics in biology—understanding those curves can really help you navigate the twists and turns of life!
