Question
Find the \( x \) and \( y \) coordinates of all infloction points. \( f(x)=x^{3}+33 x^{2} \) What islare the inflection poinl(s)? Seloct the correct choce below end, if necossary, fill in the answer box to complell your choico, A. The inflection point(s) isiare (Type an ordered pair. Use a comma to soparalo answers as neodod.) E. There are no inflection points.
Ask by Medina Cummings. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The inflection point is at \((-11, 2662)\).
Solution
**Step 1. Compute the second derivative**
Given
\[
f(x) = x^3 + 33x^2,
\]
first find the first derivative:
\[
f'(x) = 3x^2 + 66x.
\]
Then compute the second derivative:
\[
f''(x) = 6x + 66.
\]
**Step 2. Set the second derivative equal to zero**
Set
\[
6x + 66 = 0.
\]
Solve for \( x \):
\[
6x = -66 \quad \Longrightarrow \quad x = -11.
\]
**Step 3. Verify the change in concavity**
Since the second derivative is a linear function \( f''(x)=6x+66 \), it will change sign at \( x=-11 \). Hence, there is an inflection point at \( x = -11 \).
**Step 4. Find the \( y \)-coordinate of the inflection point**
Substitute \( x = -11 \) into the original function:
\[
f(-11) = (-11)^3 + 33(-11)^2.
\]
Calculate each term:
\[
(-11)^3 = -1331,
\]
\[
(-11)^2 = 121 \quad \Longrightarrow \quad 33 \times 121 = 3993.
\]
Thus,
\[
f(-11) = -1331 + 3993 = 2662.
\]
**Answer**
The inflection point is at \(\boxed{(-11,2662)}\).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
To find the inflection points of the function \( f(x) = x^3 + 33x^2 \), we need to determine where the second derivative changes sign. 1. First, find the first derivative: \( f'(x) = 3x^2 + 66x \). 2. Next, find the second derivative: \( f''(x) = 6x + 66 \). 3. Set the second derivative equal to zero to find potential inflection points: \( 6x + 66 = 0 \) \( 6x = -66 \) \( x = -11 \). 4. To find the corresponding \( y \) coordinate, substitute \( x = -11 \) back into the original function: \( f(-11) = (-11)^3 + 33(-11)^2 = -1331 + 399 = -932 \). Thus, the inflection point is at \( (-11, -932) \). A. The inflection point(s) are (-11, -932).
