JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Consider the following three statements for the function \( f:(0,\infty)\rightarrow\mathbb{R} \) 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 All (I), (II) and (III) are TRUE.
- B Only (I) is TRUE.
- C Only (II) and (III) are TRUE.
- D Only (I) and (III) are TRUE.
Answer & Solution
Correct Answer
(D) Only (I) and (III) are TRUE.
Step-by-step Solution
Detailed explanation
\(f(x)=|\ \ell n x|-|x-1|\) \(=\left\{\begin{array}{cc}\ \ell n x -( x -1) & x \geq 1 \\ -\ \ell n x +( x -1) & 0< x <1\end{array}\right.\) \(=\left\{\begin{array}{cc}\ \ell n x - x +1 & x \geq 1 \\ -\ \ell n x + x -1 & 0< x <1\end{array}\right.\)…
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