Consider the following three statements for the function � defined by � : (I) � is differentiable at all �. (II) � is increasing in �. (III) � is decreasing in �.

Consider the following three statements for the function f(x)=|logex|−|x−1|

Q. Consider the following three statements for the function f : (0, ∞) → ℝ defined by

f(x) = |log_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,

(A) Only (I) is TRUE.

(B) Only (I) and (III) are TRUE.

Correct Answer: Only (I) and (III) are TRUE.

Explanation

The given function is:

f(x) = |ln x| − |x − 1|

Consider different intervals.

For 0 < x < 1 :

|ln x| = −ln x, |x − 1| = 1 − x

So,

f(x) = −ln x − (1 − x) = x − ln x − 1

Differentiating,

f′(x) = 1 − 1/x < 0 \quad (0 < x < 1)

Hence, f is decreasing in (0,1). Statement (II) is FALSE.

For x > 1 :

|ln x| = ln x, |x − 1| = x − 1

So,

f(x) = ln x − (x − 1)

Differentiating,

f′(x) = 1/x − 1 < 0 \quad (x > 1)

Hence, f is decreasing in (1, ∞). Statement (III) is TRUE.

At x = 1, both |ln x| and |x − 1| are differentiable, hence f is differentiable for all x > 0.

Thus, statement (I) is TRUE.

Therefore, only statements (I) and (III) are true.

Explanation

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