Given the cubic function $f(x) = ax^3 + bx^2 + cx + d$ with three real roots $x_0$, $x_1$, and $x_2$ such that $x_0 < x_1 < x_2$, we need 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: Differentiate the function

To find the critical points, we first compute the derivative of the function $f(x)$.

$f'(x) = \frac{d}{dx} \left( ax^3 + bx^2 + cx + d \right) = 3ax^2 + 2bx + c$

Step 2: Analyze the roots of the derivative

The critical points of $f(x)$ occur where $f'(x) = 0$. So, we need to solve the quadratic equation:

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

This quadratic equation has at most two real roots, which we denote as $r_1$ and $r_2$. Since $a$ is non-zero, $f(x)$ is a cubic function and will have a turning point at each of these roots (assuming they are real and distinct).

Step 3: Apply the Intermediate Value Theorem

We know that the cubic function $f(x)$ has three distinct real roots $x_0$, $x_1$, and $x_2$. Let's examine the behavior of $f(x)$ near these roots:

• Since $f(x_0) = 0$, $f(x_1) = 0$, and $f(x_2) = 0$, the function must change signs between these roots.
• Between $x_0$ and $x_1$, $f(x)$ changes from negative to positive (or vice versa).
• Similarly, between $x_1$ and $x_2$, $f(x)$ changes from positive to negative (or vice versa).

Because $f(x)$ is continuous, by the Intermediate Value Theorem, there must be a point where $f'(x) = 0$ between each of these intervals, leading to critical points between $x_0$ and $x_1$, and between $x_1$ and $x_2$.

Step 4: Nature of the critical points

Now, let’s determine the nature of these critical points.

• Between $x_0$ and $x_1$, if $f(x)$ changes from increasing to decreasing, there is a local maximum at $r_1$.
• Between $x_1$ and $x_2$, if $f(x)$ changes from decreasing to increasing, there is a local minimum at $r_2$.

Given that $f(x)$ is a cubic polynomial, and the leading coefficient $a$ dictates the general behavior of the function, we can expect:

• The derivative $f'(x) = 3ax^2 + 2bx + c$ will have two real roots (assuming $b^2 - 3ac > 0$).
• The sign of $a$ will dictate the concavity of the function, ensuring that there is a local maximum in the interval $(x_0, x_1)$ and a local minimum in the interval $(x_1, x_2)$.

Conclusion

Hence, 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 more details on any part of the proof or have any questions? Here are some related questions for further exploration:

1. How would the behavior change if one of the roots is repeated?
2. What would happen if $f(x)$ had a double root?
3. How do the coefficients $a$, $b$, $c$, and $d$ affect the location of the critical points?
4. Can you graphically interpret the behavior of the cubic function around its roots?
5. How does the discriminant of the quadratic equation $3ax^2 + 2bx + c = 0$ relate to the nature of critical points?

Tip: Always consider the derivative's roots when analyzing the behavior of polynomial functions, as they provide key information about the function's turning points.