Consider the following three statements for the function f:(0, ) R defined by f(x)= | _ e x |-|x-1| : (I) f is differentiable at all x0. (II) f is increasing in (0,1). (III) f is decreasing in (1, ). Then.

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Consider the following three statements for the function f:(0, ) R defined by f(x)= | _ e x |-|x-1| : (I) f is differentiable at all x>0. (II) f is increasing in (0,1). (III) f is decreasing in (1, ). Then.

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Correct answer: D. Only (I) and (III) are TRUE.

For 0 For x > 1: f(x) = x - x + 1, f'(x) = 1 x - 1 = 1-x x At x = 1: LHD = 1-1 1 = 0, RHD = 1-1 1 = 0. So f is differentiable at x = 1. (I) TRUE: f is differentiable for all x > 0. (II) FALSE: f is decreasing in (0, 1). (III) TRUE: f is decreasing in (1, ). Only (I) and (III) are TRUE.

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