F is a continuous and monotonic function and f(-1)=10 and f(7)=2… Does f(x^3)=x had only one root in (-1,2)

We are given that \( f \) is a continuous and monotonic function, with the values \( f(-1) = 10 \) and \( f(7) = 2 \), and we are asked whether the equation \( f(x^3) = x \) has only one root in the interval \( (-1, 2) \).

### Step 1: Understanding the Problem
We need to investigate if the equation \( f(x^3) = x \) has exactly one solution in the interval \( (-1, 2) \).

### Step 2: Behavior of \( f \)
Since \( f \) is continuous and monotonic, it must either be strictly increasing or strictly decreasing over the interval in question. Let’s consider these cases separately:

1. **Case 1: \( f \) is strictly decreasing**  
   In this case, as \( x \) increases, \( f(x^3) \) decreases, and we compare it to \( x \) (which is strictly increasing).

   - At \( x = -1 \), we have \( f((-1)^3) = f(-1) = 10 \), while \( x = -1 \).
   - At \( x = 2 \), we have \( f(2^3) = f(8) \), but we are not explicitly given \( f(8) \), although \( f(7) = 2 \) suggests that \( f(8) \) should be less than or equal to 2 (since \( f \) is decreasing).
   
   The key point here is that \( f(x^3) \) is decreasing, while \( x \) is increasing, which suggests that the function \( f(x^3) - x \) changes sign and hence must have exactly one root by the **Intermediate Value Theorem**.

2. **Case 2: \( f \) is strictly increasing**  
   In this case, \( f(x^3) \) would increase as \( x \) increases. However, for the equation \( f(x^3) = x \) to have a root, we would need \( f(x^3) = x \) at some point. Given that \( f(-1) = 10 \) and \( f(7) = 2 \), this seems less likely in this case.

### Step 3: Conclusion
From the provided information, it is most consistent to assume that \( f \) is strictly decreasing. Since \( f(x^3) \) is decreasing and \( x \) is increasing, there should be exactly one point where \( f(x^3) = x \), meaning that the equation has **exactly one root** in the interval \( (-1, 2) \).

Would you like more details, or do you have any questions?

### Further Questions:
1. What happens to the number of roots if \( f \) is increasing instead of decreasing?
2. How does the monotonicity of a function influence the number of solutions?
3. Can the Intermediate Value Theorem be applied for functions that are not monotonic?
4. How would the result change if the interval was \( (-2, 2) \)?
5. What is the significance of \( f \) being continuous in this context?

### Tip:
The Intermediate Value Theorem is a powerful tool for determining the existence of roots when dealing with continuous functions that change sign over an interval.

F is a continuous and monotonic function and f(-1)=10 and f(7)=2… Does f(x^3)=x have only one root in (-1,2)?

Explore how to determine if the equation f(x^3) = x has exactly one root in the interval (-1, 2). Learn about continuous and monotonic functions and the application of the Intermediate Value Theorem in this calculus problem.

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