A Parallel Secant

★ 7/10functions calculus Substitution Monotonicity inequality Symmetry

Problem

The graph of

f(x) = \frac{\ln x}{x} + \frac{1}{x} + x

contains three points $M, N, P$ with $x$ -coordinates $x_1, x_2, 1$ respectively.

1. Find the equation of the tangent line $l$ to $f(x)$ at $P$ . 2. If the line $MN$ is parallel to $l$ , prove that

2 < \frac{1}{x_1} + \frac{1}{x_2} < \mathrm{e}.

Answer

Solution

Difficulty	7/10
Topics	functions, calculus, Substitution, Monotonicity, inequality, Symmetry

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