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For the function \(f(x)=|x-2|+|x-5|\), where \(x\in \mathbb{R}\), consider the following statements: Statement-1: \(f'(4)=0\) Statement-2: \(f\) is continuous on \([2,5]\), differentiable on \((2,5)\), and \(f(2)=f(5)\).

1. Statement-1 is false, while Statement-2 is true.
2. Both Statement-1 and Statement-2 are true, and Statement-2 correctly explains Statement-1.
3. Both Statement-1 and Statement-2 are true, but Statement-2 does not correctly explain Statement-1.
4. Statement-1 is true, while Statement-2 is false.

Correct answer: Both Statement-1 and Statement-2 are true, but Statement-2 does not correctly explain Statement-1.

Solution

Statement-1 is true because the function has a minimum at x=4, where the derivative is zero. Statement-2 is also true as the function is continuous and differentiable in the specified intervals, but it does not provide a reason for the derivative being zero at that specific point.

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