Rolle's Theorem can be used to analyze the existence of multiple real solutions to an equation involving continuous functions, but it is specific to differentiable functions over a closed interval. Let's review Rolle's Theorem and see how it applies here.

### Rolle's Theorem

Rolle's Theorem states that if a function \( f \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and if \( f(a) = f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

### Application to the Given Problem

For the equation \( f(x) = m \) to have exactly three real solutions, consider the function \( f(x) = x^3 \). We want to find the values of \( m \) for which the equation \( x^3 = m \) has exactly three real solutions. 

Let's consider the polynomial function \( f(x) = x^3 + ax^2 + bx + c \), which is a general cubic function. The number of real roots of this equation depends on the nature of its critical points and how many times the graph intersects the horizontal line \( y = m \).

1. **Finding Critical Points:**
   The critical points of \( f(x) \) are found by setting its derivative to zero:
   \[ f'(x) = 3x^2 + 2ax + b \]
   Solving \( f'(x) = 0 \) gives:
   \[ 3x^2 + 2ax + b = 0 \]

2. **Discriminant of the Derivative:**
   The quadratic equation \( 3x^2 + 2ax + b = 0 \) has two distinct real roots if its discriminant is positive:
   \[ (2a)^2 - 4 \cdot 3 \cdot b > 0 \]
   \[ 4a^2 - 12b > 0 \]
   \[ a^2 > 3b \]

If \( a^2 > 3b \), then \( f(x) \) has two critical points. By Rolle's Theorem, if a continuous and differentiable function has two critical points, there exists at least one point between them where the derivative is zero, which implies that the function changes direction.

3. **Number of Real Solutions:**
   - If the function changes direction twice (i.e., has a local maximum and a local minimum), and if \( f(x) = m \) intersects the horizontal line \( y = m \) three times, then there are three real solutions.
   - For \( f(x) = x^3 \), we don't have the quadratic term (\(ax^2\)) and linear term (\(bx\)), making the analysis simpler. It has a single inflection point at \( x = 0 \), and thus can only intersect a horizontal line at most three times if the horizontal line passes through the origin, i.e., \( m = 0 \).

For a general cubic polynomial \( f(x) = x^3 + ax^2 + bx + c \), the condition for exactly three real roots is determined by the sign of the discriminant and how the function intersects the horizontal line \( y = m \).

### Summary

For the specific function \( f(x) = x^3 \), Rolle's Theorem does not directly apply since the function does not have local extrema. However, for a more general cubic polynomial, Rolle's Theorem helps to identify critical points and analyze the behavior of the function, leading to the determination of how many real roots it can have.

Would you like more details or have any questions about these explanations? Here are 5 additional questions you might consider:

1. How do critical points influence the shape of a cubic function?
2. Can you explain the discriminant and its role in determining the nature of roots?
3. How does the Intermediate Value Theorem apply to polynomial equations?
4. What are the conditions for a cubic equation to have exactly three real roots?
5. How do inflection points differ from local maxima and minima?

**Tip:** When dealing with polynomial equations, always examine the critical points and the sign of the discriminant to determine the nature and number of real roots.

Explore how Rolle's Theorem applies to cubic functions in mathematics. Learn about critical points, discriminants, and their roles in determining the number of real roots of polynomial equations. Suitable for advanced high school students and those interested in calculus.

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