Given the function $f(x) = ax^3 + bx^2 + cx + d$, with real roots $x_0 < x_1 < x_2$, we want to show that there must be a local maximum between $x_0$ and $x_1$, and a local minimum between $x_1$ and $x_2$.

Step 1: Finding the Derivative

The first step is to find the critical points of the function, which occur where the derivative is zero. The derivative of the function is:

$f'(x) = 3ax^2 + 2bx + c.$

Step 2: Identifying Critical Points

The critical points occur when $f'(x) = 0$. To find these points, solve the quadratic equation:

$3ax^2 + 2bx + c = 0.$

Since the function $f(x)$ is a cubic polynomial, its derivative $f'(x)$ is a quadratic function, which can have at most two real roots. These roots represent the critical points, where the function could have local maxima or minima.

Let the roots of $f'(x) = 0$ be denoted by $x_3$ and $x_4$, such that $x_3 < x_4$. These are the locations where the slope of the function is zero, indicating potential local extrema.

Step 3: Analyzing the Behavior of the Function

Given that $f(x)$ has three real roots $x_0 < x_1 < x_2$, the behavior of the cubic function changes sign at each of these roots:

• For $x < x_0$, $f(x)$ is positive or negative depending on the leading coefficient $a$.
• $f(x)$ changes sign at each root.

Now, consider the intervals between these roots:

1. Interval $(x_0, x_1)$:

   Since $f(x)$ changes from positive to negative (or vice versa), there must be a turning point between $x_0$ and $x_1$. This turning point corresponds to a critical point $x_3$ between $x_0$ and $x_1$. Since the function moves from increasing to decreasing, $x_3$ must be a local maximum.

2. Interval $(x_1, x_2)$:

   Similarly, since $f(x)$ again changes from positive to negative (or vice versa), there must be another turning point between $x_1$ and $x_2$. This corresponds to a critical point $x_4$ between $x_1$ and $x_2$. Here, the function moves from decreasing to increasing, so $x_4$ must be a local minimum.

Step 4: Conclusion

Thus, there must be:

• A local maximum between $x_0$ and $x_1$ at $x_3$.
• A local minimum between $x_1$ and $x_2$ at $x_4$.

This is guaranteed by the nature of the cubic function, which has inflection points corresponding to the changes in concavity dictated by its critical points.

Would you like more details or have any questions?

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Related Questions:

1. How do the coefficients $a, b, c,$ and $d$ affect the location of the extrema?
2. What happens to the local extrema if the function has a double root?
3. How do you determine if a critical point is a maximum, minimum, or inflection point?
4. Can a cubic function have more than two critical points?
5. What is the role of the discriminant of the derivative in determining the nature of the extrema?

Tip: For cubic functions, always check the derivative to understand the location and nature of critical points.