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 $x_2$.

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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.