A Unique Separating $k$

Problem

Let $f(x) = kx^2 - kx$ and

g(x) = \begin{cases} \ln x, & x \geqslant 1,\\ -x^3 + (a+1)x^2 - ax, & 0 < x < 1. \end{cases}

Suppose the real number $k$ making $f(x) \geqslant g(x)$ hold for all positive $x$ exists and is unique. Find the value of the real number $a$ .

Answer

Solution

Difficulty	8/10
Topics	functions, calculus, Monotonicity

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