A Shifted Extreme Point

★ 8/10functions Extrema calculus Monotonicity Logarithms

Problem

Let $f(x) = x - \dfrac12\sin x - \ln x$ .

1. Find the maximum of $f$ on $\left[1, \dfrac{\pi}{3}\right]$ . 2. Prove that $f'$ has a unique zero. 3. If $x_1 \ne x_2$ and $f(x_1) = f(x_2)$ , prove $x_1x_2 < 4$ .

Answer

Solution

Difficulty	8/10
Topics	functions, Extrema, calculus, Monotonicity, Logarithms

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