For the function \( f(x)=-2\left|\log_e|x|\right| \), which of the following is true?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → It is continuous and differentiable for all \( x\gt 1 \).
The function is defined for every x ≠ 0. For x > 1, \( \log_e x\gt 0 \), so \( f(x)=-2\log_e x \); for 0 < x < 1, \( \log_e x\lt 0 \), so \( f(x)=2\log_e x \).
Option (1) It is continuous and differentiable for all \( x\gt 1 \). — Correct. For x > 1, \( f(x)=-2\log_e x \), which is continuous and differentiable.
Option (2) \( f(5)\gt 0 \) — Incorrect. \( f(5)=-2\log_e 5\lt 0 \).
Option (3) It is differentiable at \( x=1 \). — Incorrect. Just to the right of 1 the slope is −2 and just to the left it is +2, so the graph has a corner at x = 1.
Option (4) It is continuous and differentiable at \( x=0 \). — Incorrect. f is not even defined at x = 0, since \( \log_e|x| \) needs x ≠ 0.