If $|\ln x| = px$ has exactly three distinct solutions, then find $[p]$ (where $[.]$ denote greater integer function).
Step-by-Step Solution
Key Concept: The equation |ln x| = px has exactly three solutions when the line y = px is tangent to y = ln x (for x > 1) at exactly one point. This tangency condition requires finding where the derivative of ln x equals p, and the resulting p value must satisfy 0 < p < 1/e for three intersection points total (one in 0 < x < 1 region, one at tangent point, one in x > 1 region).
The line $y = px$ is tangent to $y = |\ln x|$ when it touches the curve. For the tangent line to have exactly three intersections with $y = |\ln x|$, we require the slope $p$ to be in the range $\left(0, \frac{1}{e}\right)$. At the critical value $p = \frac{1}{e}$, the line touches $y = \ln x$ at $x = e$. Therefore $[p] = 0$.
Correct Answer: 0