Question
Identify the intervals on which the function
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Answer
The function
Solution
To determine the intervals on which the function
Step 1: Find the derivative
Using the product rule, we have:
Calculating the derivatives:
Now substituting these into the product rule:
This simplifies to:
Step 2: Simplify
Now we will simplify
Expanding both terms:
Combining these:
Step 3: Factor
Factoring out the common terms:
Step 4: Find critical points
To find where
This gives us the critical points:
Step 5: Analyze the sign of
We will test the intervals determined by the critical points
-
Interval
: - Choose
:
(Increasing) - Choose
-
Interval
: - Choose
:
(Decreasing) - Choose
-
Interval
: - Choose
:
(Increasing) - Choose
Conclusion
The function
Now, let’s check the provided intervals:
-
: Increasing (correct) -
: Increasing (correct) -
: Increasing (correct)
Thus, the correct intervals where the function is always increasing are:
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Simplify this solution Beyond the Answer
To find the intervals where the function
Calculating the derivative, we find critical points that help identify test intervals. By testing the sign of the derivative in those intervals, we discover that the function increases on
When looking deeper into the function’s behavior, it’s fascinating how polynomial functions like this one can feature local maxima and minima. Knowing how to differentiate and analyze the critical points really showcases the beautiful relationship between calculus and polynomial characteristics.
If you’re curious about diving into more complex polynomial behaviors, you might enjoy exploring topics like end behavior and the Fundamental Theorem of Algebra. Understanding these concepts can provide insight into how polynomials behave overall, beyond just their increasing or decreasing intervals!
