A Separating Line

★ 6/10functions calculus Tangent Line Trick Logarithms

Problem

For functions $f(x), g(x)$ with common domain $D$ , a line $y = ax + b$ is a separating line if

f(x) \geqslant ax + b \geqslant g(x) \quad \text{for all } x \in D.

Let $f(x) = x^2 + x - x\ln x - 3$ and $g(x) = -x^2 + 4x - 4$ , both for $x \geqslant 1$ . Find the equation of the separating line of the curves $y = f(x)$ and $y = g(x)$ .

Answer

Solution

Difficulty	6/10
Topics	functions, calculus, Tangent Line Trick, Logarithms

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