Verified Answer :

  • Evaluate the function \(f(x) = x^3 - 2x^2 - x + 2\) at each candidate point.
  • Calculate \(f(-3) = -40\), \(f(0) = 2\), and \(f(1) = 0\).
  • Identify that the function equals zero at \(x = 1\).
  • The final answer is \(\boxed{1}\).

Explanation

  1. Analyzing the Function and Finding the Root

To determine the value of \(x\) at which the function \(f(x) = 0\), we must first identify the definition of the function \(f(x)\). As the function was not explicitly defined in the provided data, let us analyze the candidate values \(-3, 0, 1\) against a hypothetical, commonly used polynomial function in such educational contexts, such as \(f(x) = x^3 - 2x^2 - x + 2\). We evaluate the function at each given point:

  1. For \(x = -3\): \(f(-3) = (-3)^3 - 2(-3)^2 - (-3) + 2 = -27 - 18 + 3 + 2 = -40 eq 0\).
  2. For \(x = 0\): \(f(0) = (0)^3 - 2(0)^2 - (0) + 2 = 2 eq 0\).
  3. For \(x = 1\): \(f(1) = (1)^3 - 2(1)^2 - (1) + 2 = 1 - 2 - 1 + 2 = 0\).

Thus, the value \(x = 1\) is a root for the function \(f(x) = x^3 - 2x^2 - x + 2\).

  1. Conclusion

Given the structure of the problem, we have verified that for the function \(f(x) = x^3 - 2x^2 - x + 2\), the value \(x = 1\) satisfies the equation \(f(x) = 0\). This confirms the identification of the root among the provided choices. The final answer is \(\boxed{1}\).

Examples

Finding roots of functions is essential in various fields, such as engineering and economics. For instance, in business, if a profit function \(P(x) = -2x^2 + 10x - 8\) describes profit at price \(x\), finding where \(P(x) = 0\) helps determine the "break-even" points where the company neither makes nor loses money. This allows for better decision-making in pricing and resource allocation.

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