A Max of Three Ratios

★ 7/10functions Extrema algebra inequality Discriminant

Problem

Let $a \geqslant b \geqslant c > 0$ , and suppose the function

f(x) = \ln(2ax^2 - bx + c)

has range $\mathbb{R}$ . Find the minimum value of

\max\left\{\frac{a}{b+c},\ \frac{b}{c+a},\ \frac{c}{a+b}\right\}.

Answer

Solution

Difficulty	7/10
Topics	functions, Extrema, algebra, inequality, Discriminant

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