Consider a function f(x) = ax3 + bx2 + cx + d, with three real roots
x0, x1, x2 such that x0 < x1 < x2. Show that there must be a local maximum
between x0 and x1 and a local minimum between x1 and x2.

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 \( x_2 \).

---

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.

Explore how cubic functions with three real roots exhibit local extrema. Learn about the derivative of cubic functions and apply the Intermediate Value Theorem to prove the existence of local maxima and minima.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation