Question

Let $f(x)=|\log_e x|-|x-1|+5$. Statement 1: $f(x)$ is differentiable for all $x\in(0,\infty)$
Statement 2: $f(x)$ is increasing in $(1,\infty)$
Statement 3: $f(x)$ is decreasing in $(0,1)$
Which of the following is correct?

• A. Statement 1
• B. Statement 2
• C. Statement 3
• D. Statement 4

Hint:

Always remove modulus by breaking the function into intervals before checking monotonicity or differentiability.

Answer 1

The Correct Option is A, C

Step 1: Given function

f(x) = |ln x| − |x − 1| + 5

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Step 2: Check differentiability

The function |ln x| is differentiable for x > 0 except at x = 1, where ln x = 0.

The function |x − 1| is differentiable for all x > 0 except at x = 1.

Since both absolute value terms are non-differentiable at x = 1, the function f(x) is not differentiable at x = 1.

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Step 3: Monotonicity for x > 1

For x > 1:

|ln x| = ln x,   |x − 1| = x − 1

f(x) = ln x − (x − 1) + 5

f′(x) = 1/x − 1

For increasing behavior:

1/x − 1 > 0 ⇒ x < 1

This condition is not satisfied for x > 1.

Hence, f(x) is not increasing in (1, ∞).

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Step 4: Monotonicity for 0 < x < 1

For 0 < x < 1:

|ln x| = −ln x,   |x − 1| = 1 − x

f(x) = −ln x − (1 − x) + 5

f′(x) = −1/x + 1

For decreasing behavior:

−1/x + 1 < 0 ⇒ x < 1

This condition holds for all x ∈ (0, 1).

Hence, f(x) is decreasing in (0, 1).

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Final Conclusion:

✔ The function is not differentiable at x = 1
✔ The function is decreasing in (0, 1)

Correct statements: Statement 1 and Statement 3