The coordinates of the local maximum and minimum turning points of f(x) = x³ + 3x² are:
 A. (0 ; 0) and (-3 ; 0)
 B. (0 ; 0) and (4 ; -2)
 C. none of the other options
 D. (0 ; 0) and (-2 ; 4)

To find the coordinates of the local maximum and minimum turning points for the function \( f(x) = x^3 + 3x^2 \), follow these steps:

1. **Find the first derivative** to determine the critical points:
   \[
   f(x) = x^3 + 3x^2
   \]
   Differentiate the function with respect to \( x \):
   \[
   f'(x) = 3x^2 + 6x
   \]
   Factor the derivative:
   \[
   f'(x) = 3x(x + 2)
   \]

2. **Set the first derivative equal to zero** to find the critical points:
   \[
   3x(x + 2) = 0
   \]
   This gives us two critical points:
   \[
   x = 0 \quad \text{and} \quad x = -2
   \]

3. **Find the corresponding \( y \)-coordinates** by substituting these \( x \)-values back into the original function \( f(x) \):
   - For \( x = 0 \):
     \[
     f(0) = 0^3 + 3(0)^2 = 0
     \]
     The point is \( (0, 0) \).
   - For \( x = -2 \):
     \[
     f(-2) = (-2)^3 + 3(-2)^2 = -8 + 12 = 4
     \]
     The point is \( (-2, 4) \).

4. **Determine the nature of these critical points** (whether they are maximum or minimum) by checking the sign of the second derivative:
   \[
   f''(x) = 6x + 6
   \]
   - For \( x = 0 \):
     \[
     f''(0) = 6(0) + 6 = 6 \quad (\text{positive, so } (0, 0) \text{ is a local minimum}).
     \]
   - For \( x = -2 \):
     \[
     f''(-2) = 6(-2) + 6 = -12 + 6 = -6 \quad (\text{negative, so } (-2, 4) \text{ is a local maximum}).
     \]

Thus, the coordinates of the local maximum and minimum turning points are **D. (0, 0) and (-2, 4)**.

Would you like more details or have any questions? Here are some related questions:

1. How do you apply the second derivative test to classify critical points?
2. What does it mean for a point to be a "turning point" in the context of calculus?
3. How can you determine if a function has inflection points?
4. Can a function have a local minimum or maximum without a turning point?
5. How does the sign of the first derivative relate to the increasing or decreasing behavior of a function?

**Tip:** Always analyze both the first and second derivatives when identifying and classifying turning points, as this ensures accurate determination of their nature (maximum, minimum, or neither).

Discover how to find the local maximum and minimum turning points of the function f(x) = x³ + 3x². Step-by-step guide including differentiation, critical point analysis, and second derivative test. Suitable for Grades 11-12 students studying calculus.

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