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Statement -1: f(x) = x^(3)-3x+1 =0 has o...

Statement -1: `f(x) = x^(3)-3x+1 =0` has one root in the interval [-2,2].
Statement-2: `f(-2) and f(2) are of opposite sign.

A

Statement-1 : is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1

B

Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanantion for Statement-1

C

Statement-1 is True, Statement-2 is False

D

Statement -1 is False, Statement-2 is False

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the two statements given regarding the function \( f(x) = x^3 - 3x + 1 \). ### Step 1: Evaluate \( f(-2) \) We start by calculating \( f(-2) \): \[ f(-2) = (-2)^3 - 3(-2) + 1 \] \[ = -8 + 6 + 1 \] \[ = -8 + 7 = -1 \] ### Step 2: Evaluate \( f(2) \) Next, we calculate \( f(2) \): \[ f(2) = (2)^3 - 3(2) + 1 \] \[ = 8 - 6 + 1 \] \[ = 8 - 5 = 3 \] ### Step 3: Analyze the signs of \( f(-2) \) and \( f(2) \) Now we check the signs of \( f(-2) \) and \( f(2) \): - \( f(-2) = -1 \) (which is negative) - \( f(2) = 3 \) (which is positive) Since \( f(-2) \) and \( f(2) \) are of opposite signs, we can apply the Intermediate Value Theorem. ### Step 4: Apply the Intermediate Value Theorem According to the Intermediate Value Theorem, if a continuous function changes signs over an interval, then there exists at least one root in that interval. Since \( f(x) \) is a polynomial (and hence continuous), and we have: - \( f(-2) < 0 \) - \( f(2) > 0 \) We conclude that there is at least one root in the interval \([-2, 2]\). ### Conclusion Thus, we can confirm that: - Statement 1 is true: \( f(x) = x^3 - 3x + 1 = 0 \) has one root in the interval \([-2, 2]\). - Statement 2 is also true: \( f(-2) \) and \( f(2) \) are of opposite signs. ### Final Answer Both statements are true, and Statement 2 correctly explains Statement 1. ---
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