Two Tangents Meet Below

★ 7/10functions Log Mean Inequality calculus Tangent Line Trick Monotonicity

Problem

The line $y = kx$ meets the curve $y = \ln x$ at two distinct points $M(x_1, y_1)$ and $N(x_2, y_2)$ , and the tangent lines to $y = \ln x$ at $M$ and $N$ meet at $P(x_0, y_0)$ . Determine, with proof, which of the following are true:

1. $0 < k < \dfrac{1}{\mathrm{e}}$ ; 2. $x_1x_2 = \mathrm{e}x_0$ ; 3. $y_1 + y_2 = 1 + y_0$ ; 4. $y_1y_2 < 1$ .

Answer

Solution

Difficulty	7/10
Topics	functions, Log Mean Inequality, calculus, Tangent Line Trick, Monotonicity

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