Given the function $f(x) = ax^3 + bx^2 + cx + d$ with real roots $x_0$, $x_1$, and $x_2$ such that $x_0 < x_1 < x_2$, we are asked 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: Analyze the first derivative of $f(x)$

The first step is to find the critical points of $f(x)$ by taking its first derivative:

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

The critical points occur where $f'(x) = 0$, i.e.,

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

This is a quadratic equation, and its roots (let's call them $r_1$ and $r_2$) can be found using the quadratic formula:

$r_1, r_2 = \frac{-2b \pm \sqrt{(2b)^2 - 4(3a)(c)}}{2(3a)} = \frac{-2b \pm \sqrt{4b^2 - 12ac}}{6a} = \frac{-b \pm \sqrt{b^2 - 3ac}}{3a}$

Step 2: Understand the significance of the critical points $r_1$ and $r_2$

The roots $r_1$ and $r_2$ are the critical points where $f(x)$ could have local extrema (maximum or minimum). Since $f(x)$ is a cubic polynomial, it will have at most two real critical points.

Step 3: Location of the critical points relative to the roots of $f(x)$

Given that the function $f(x)$ has three distinct real roots $x_0$, $x_1$, and $x_2$ (with $x_0 < x_1 < x_2$), the following must hold:

• Between $x_0$ and $x_1$, $f(x)$ changes from positive to negative (since $f(x_0) = 0$ and $f(x_1) = 0$).
• Between $x_1$ and $x_2$, $f(x)$ changes from negative to positive (since $f(x_1) = 0$ and $f(x_2) = 0$).

For $f(x)$ to have such changes in sign between the roots, $f'(x)$ must change sign, which implies that there is a critical point in each of these intervals:

1. There must be a critical point $r_1$ between $x_0$ and $x_1$, where $f'(x) = 0$ and $f'(x)$ changes from positive to negative, indicating a local maximum.

2. There must be a critical point $r_2$ between $x_1$ and $x_2$, where $f'(x) = 0$ and $f'(x)$ changes from negative to positive, indicating a local minimum.

Step 4: Conclusion

Therefore, there must be a local maximum between $x_0$ and $x_1$, and a local minimum between $x_1$ and $x_2$. This is a consequence of the Intermediate Value Theorem, which ensures that the continuous derivative $f'(x)$ must cross zero in these intervals, combined with the fact that $f(x)$ changes sign between the roots.

Would you like further details or have any questions?

Here are some related questions you might find useful:

1. What is the significance of the discriminant in determining the nature of the roots for cubic functions?
2. How do the roots of the first derivative relate to the concavity of the original function?
3. Can a cubic function have more than one local maximum or minimum?
4. How can you determine the inflection point of a cubic function?
5. What happens to the nature of the critical points if the roots of the cubic function are not distinct?

Tip: When analyzing cubic functions, always check the first derivative to find critical points and the second derivative to determine the nature (max/min) of these points.