Where is the point of infection for the function y = x3 + 6x2?

To find the point of inflection for the function \( y = x^3 + 6x^2 \), we need to follow these steps:

1. **Find the first derivative** of the function:
   \[
   y' = \frac{d}{dx}(x^3 + 6x^2) = 3x^2 + 12x
   \]

2. **Find the second derivative** of the function:
   \[
   y'' = \frac{d}{dx}(3x^2 + 12x) = 6x + 12
   \]

3. **Set the second derivative equal to zero** and solve for \( x \) to find potential points of inflection:
   \[
   6x + 12 = 0
   \]
   \[
   6x = -12
   \]
   \[
   x = -2
   \]

4. **Verify the change in concavity** around \( x = -2 \) by checking the sign of \( y'' \) on either side of \( x = -2 \):
   - For \( x < -2 \) (e.g., \( x = -3 \)):
     \[
     y''(-3) = 6(-3) + 12 = -18 + 12 = -6 \quad (\text{negative})
     \]
   - For \( x > -2 \) (e.g., \( x = -1 \)):
     \[
     y''(-1) = 6(-1) + 12 = -6 + 12 = 6 \quad (\text{positive})
     \]

Since \( y'' \) changes from negative to positive at \( x = -2 \), there is a point of inflection at \( x = -2 \).

5. **Find the corresponding \( y \)-value**:
   \[
   y(-2) = (-2)^3 + 6(-2)^2 = -8 + 6(4) = -8 + 24 = 16
   \]

So, the point of inflection for the function \( y = x^3 + 6x^2 \) is at \((-2, 16)\).

Would you like more details or have any questions?

Here are some questions you might ask next:
1. How do you determine the concavity of a function?
2. Can you explain why the second derivative test works for finding points of inflection?
3. How can you find inflection points for more complex functions?
4. What is the significance of an inflection point in the graph of a function?
5. Can there be multiple points of inflection for a single function?

**Tip**: When solving for points of inflection, always verify that the second derivative changes sign at the solution point to confirm a change in concavity.

Discover how to find the point of inflection for the function y = x^3 + 6x^2 using calculus. Learn the step-by-step process of finding inflection points, verifying concavity changes, and determining corresponding y-values. Suitable for college-level calculus students.

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