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Consider the following statements:
Statement I: For the fixed point (0, y₀) with 0 < y₀ < (1)/(2), the least distance from the parabola y = x² is y₀.
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 false.
Solution
Distance^2 from (0,y0) to (x,x^2) is x^2+(x^2-y0)^2; its derivative 2x[1+2(x^2-y0)]=0 has only x=0 when y0<1/2, giving least distance y0, so I is true. Statement II is false because extrema can occur where f' does not exist (e.g. |x| at 0) or at endpoints. Hence I true, II false.
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