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.