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Consider the function f : [1/2, 1] → R defined by f(x) = 4√2 x^3 − 3√2 x − 1. Consider the statements (I) The curve y = f(x) intersects the x-axis exactly at one point. (II) The curve y = f(x) intersects the x-axis at x = cos(π/12) Then

1. Both (I) and (II) are incorrect.
2. Only (II) is correct.
3. Only (I) is correct.
4. Both (I) and (II) are correct.

Correct answer: Both (I) and (II) are correct.

Solution

Both statements are correct because the function f(x) has been analyzed and found to intersect the x-axis at exactly one point, which is at x = cos(π/12), confirming that both claims about the intersection are valid.

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