Consider a function f(x) = ax3 + bx2 + cx + d, with three real roots
x0, x1, x2 such that x0 < x1 < x2. Show that there must be a local maximum
between x0 and x1 and a local minimum between x1 and x2.

To prove that the function \( f(x) = ax^3 + bx^2 + cx + d \), which has three real roots \( x_0 < x_1 < x_2 \), must have a local maximum between \( x_0 \) and \( x_1 \), and a local minimum between \( x_1 \) and \( x_2 \), we can follow these steps:

### 1. **Find the derivative of the function**:
The critical points of the function, which correspond to the local maximum and minimum, are found where the derivative \( f'(x) \) is equal to zero.

Given \( f(x) = ax^3 + bx^2 + cx + d \), the derivative is:
\[
f'(x) = 3ax^2 + 2bx + c
\]

### 2. **Analyze the roots of the derivative**:
Since \( f(x) \) is a cubic function, its derivative \( f'(x) \) is a quadratic function. The quadratic equation \( f'(x) = 0 \) can be solved as:
\[
3ax^2 + 2bx + c = 0
\]
The roots of this quadratic equation are given by the quadratic formula:
\[
x = \frac{-2b \pm \sqrt{(2b)^2 - 4(3a)c}}{2(3a)} = \frac{-2b \pm \sqrt{4b^2 - 12ac}}{6a}
\]
\[
x = \frac{-b \pm \sqrt{b^2 - 3ac}}{3a}
\]

### 3. **Nature of the roots of \( f'(x) \)**:
The discriminant of the quadratic equation \( f'(x) = 0 \) is \( \Delta = b^2 - 3ac \).

- If \( \Delta > 0 \), there are two distinct real roots of \( f'(x) = 0 \). This means \( f(x) \) has one local maximum and one local minimum.
- If \( \Delta = 0 \), the quadratic has a double root, which corresponds to a point of inflection where \( f(x) \) does not have a distinct local maximum or minimum.
- If \( \Delta < 0 \), the quadratic has no real roots, implying that \( f'(x) = 0 \) has no real solution, and \( f(x) \) does not have any local maximum or minimum.

Given that \( f(x) \) is a cubic function with three distinct real roots \( x_0 < x_1 < x_2 \), the function must change direction between each pair of consecutive roots. Therefore, \( \Delta > 0 \) to ensure there are two distinct real roots of \( f'(x) = 0 \).

### 4. **Sign changes and Intermediate Value Theorem**:
Between any two consecutive roots of \( f(x) \), the sign of \( f(x) \) changes (since these are roots). For a cubic function, this change implies that the function must have a turning point (local maximum or minimum) between these roots.

- Between \( x_0 \) and \( x_1 \), there is a sign change in \( f(x) \), and therefore, there must be a critical point in \( (x_0, x_1) \). Since the function changes from increasing to decreasing as it passes through \( x_1 \), this critical point is a **local maximum**.
  
- Similarly, between \( x_1 \) and \( x_2 \), there is a sign change in \( f(x) \), and therefore, there must be another critical point in \( (x_1, x_2) \). As the function changes from decreasing to increasing as it passes through \( x_2 \), this critical point is a **local minimum**.

### 5. **Conclusion**:
Thus, the function \( f(x) = ax^3 + bx^2 + cx + d \) must have a local maximum between \( x_0 \) and \( x_1 \), and a local minimum between \( x_1 \) and \( x_2 \).

---

Would you like further details on any of these steps or have any questions?

Here are five related questions to expand on this topic:

1. How does the behavior of a cubic function differ when the discriminant of the derivative is zero?
2. What would happen if the roots \( x_0, x_1, x_2 \) were not distinct?
3. Can a cubic function have more than one local maximum or minimum?
4. How does the sign of the leading coefficient \( a \) affect the nature of the cubic function?
5. How can the location of the critical points be determined precisely for specific values of \( a \), \( b \), and \( c \)?

**Tip:** When analyzing polynomial functions, always start by examining the derivative to find critical points, which indicate potential maxima, minima, or points of inflection.

Learn how to prove the existence of local maximums and minimums in cubic functions with three real roots using derivative analysis and the discriminant of quadratic equations. Suitable for high school students studying advanced algebra.

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