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Consider the following statements: Statement I: For the fixed point \((0, y_0)\) with \(0 < y_0 < \tfrac{1}{2}\), the least distance from the parabola \(y = x^2\) is \(y_0\). Statement II: Every point where a function attains a maximum or minimum is necessarily a solution of \(f'(x)=0\). Choose the correct option.
- Statement I is true, Statement II is true, and Statement II correctly explains Statement I.
- Statement I is true, Statement II is true, but Statement II does not correctly explain Statement I.
- Statement I is true, Statement II is false.
- Statement I is false, Statement II is true.
Correct answer: Statement I is true, Statement II is true, but Statement II does not correctly explain Statement I.
Solution
Statement I is true because the vertical distance from the point (0, y_0) to the parabola is minimized at the point where the parabola is closest to that y-value, which occurs at y = y_0 when y_0 is less than 1/2. Statement II is also true as critical points, where a function reaches a maximum or minimum, occur where the derivative is zero, but this does not relate to the geometric context of Statement I.
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