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

Step 1: Identify the critical points of $f(x)$

To find the critical points, we first need to compute the derivative of $f(x)$:

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

The critical points occur where the derivative is zero:

$f'(x) = 0 \implies 3ax^2 + 2bx + c = 0$

Step 2: Analyze the roots of the derivative

The quadratic equation $3ax^2 + 2bx + c = 0$ can have either two real roots, one repeated real root, or no real roots. However, since the cubic function $f(x)$ has three real roots and its leading coefficient $a$ is non-zero (which means it is not a constant or linear function), the quadratic $3ax^2 + 2bx + c = 0$ must have two distinct real roots. Let's denote these roots as $r_1$ and $r_2$, with $r_1 < r_2$.

Step 3: Relationship between the roots of $f(x)$ and $f'(x)$

Since $f(x)$ is a cubic polynomial with three real roots $x_0$, $x_1$, and $x_2$, the graph of $f(x)$ will cross the x-axis three times. The nature of cubic functions is such that between any two consecutive roots $x_i$ and $x_{i+1}$, the function $f(x)$ will either have a local maximum or minimum depending on the sign of $f'(x)$.

Since $r_1$ and $r_2$ are the roots of $f'(x) = 0$, these correspond to the points where the slope of $f(x)$ is zero. These points must occur between the roots of $f(x)$ because $f(x)$ changes sign at its roots.

• Local Maximum between $x_0$ and $x_1$: Since $f(x)$ changes sign from negative to positive at $x_0$ and from positive to negative at $x_1$, there must be a local maximum at some point between $x_0$ and $x_1$, which corresponds to $r_1$.

• Local Minimum between $x_1$ and $x_2$: Similarly, since $f(x)$ changes sign from positive to negative at $x_1$ and from negative to positive at $x_2$, there must be a local minimum at some point between $x_1$ and $x_2$, which corresponds to $r_2$.

Step 4: Conclusion

Thus, we have shown that the cubic 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 $x_2$, corresponding to the critical points $r_1$ and $r_2$, respectively.

Would you like further details on this topic or any other related concepts?

Here are some related questions:

1. How do the coefficients $a$, $b$, $c$, and $d$ influence the shape of a cubic function?
2. Can you generalize this result for a polynomial of higher degree with more than three roots?
3. What are the implications of the discriminant of the quadratic derivative on the nature of critical points?
4. How does the behavior of a cubic function differ if the roots are not distinct?
5. What are the real-world applications of identifying local maxima and minima in cubic functions?

Tip: When analyzing a cubic function, understanding the relationship between its derivative and the original function can help you predict the locations of critical points and their nature.