A Mixed-Root Maximum

★ 7/10algebra Estimation AM-GM inequality

Problem

Positive reals $a, b, c$ satisfy $a + b + c = 1$ . Let $m$ be the maximum value of

a + \sqrt{b} + \sqrt[4]{c}.

Determine which of $1$ , $\dfrac{5}{4}$ , $\dfrac{3}{2}$ , $\dfrac{7}{4}$ is closest to $m$ , with proof.

Answer

Solution

Difficulty	7/10
Topics	algebra, Estimation, AM-GM, inequality

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