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.

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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