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The document you've provided includes a mathematics assignment for Grade 12, titled "Task 2024" and involves investigating the inflection point of a cubic graph. Below is a breakdown of the questions:

### **Question 1:**
- You're given the function \( f(x) = 2x^3 - 5x^2 - 4x + 3 \).
1.1. Draw the graph of \( f \), indicating all intercepts and turning points.
1.2. Determine the first derivative, \( g(x) \), of \( f(x) \).
1.3. Draw the graph of \( g(x) \), including intercepts and turning points.
1.4. Determine the second derivative, \( h(x) \), and draw it.
1.5. Investigate the relationship between the \( x \)-intercepts of \( g(x) \) and the turning points of \( f(x) \).
1.6. Determine the inflection point of \( f(x) \) using two methods.
1.7. Observe the relationship between the axis of symmetry of \( g(x) \), the \( x \)-intercept of \( h(x) \), and the \( x \)-coordinate of the inflection point of \( f(x) \).

### **Question 2:**
- You're given \( f(x) = -x^3 + 4x^2 + 11x - 30 \).
2.1. Draw the graph of \( f(x) \), indicating intercepts and turning points.
2.2. Find the first derivative, \( g(x) \), of \( f(x) \).
2.3. Graph \( g(x) \), showing intercepts and turning points.
2.4. Find the second derivative, \( h(x) \), and graph it.
2.5. Investigate the relationship between the \( x \)-intercepts of \( g(x) \) and the turning points of \( f(x) \).
2.6. Determine the inflection point of \( f(x) \).
2.7. Investigate the relationship between the axis of symmetry of \( g(x) \), the \( x \)-intercept of \( h(x) \), and the \( x \)-coordinate of the inflection point of \( f(x) \).

### **Question 3: Conclusion**
- Draw a conclusion about the relationship between the inflection point of a cubic function and the graphs of its first and second derivatives.

### **Question 4: Application**
- This involves analyzing the graph of \( g(x) \), determining where it's strictly increasing or decreasing, and classifying stationary points.

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Would you like help solving any specific parts of this investigation? Let me know!

Here are five related questions that you might explore:
1. How do you calculate the inflection point of a cubic function?
2. What is the significance of the second derivative in determining concavity?
3. How can you determine turning points using the first derivative?
4. What is the relationship between the roots of the derivative and turning points?
5. How do changes in coefficients affect the shape of a cubic function?

**Tip:** The inflection point occurs where the second derivative changes sign, indicating a change in concavity.

Investigate the inflection point of a cubic graph and its relationship to the graphs of the first and second derivatives.

Investigate the inflection point of cubic functions and understand its relationship with the graphs of the first and second derivatives. This task involves graphing, deriving, and analyzing concavity for cubic equations. Suitable for Grade 12 students.

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