A Two-Parameter Minimum

★ 8/10functions Substitution Absolute Value inequality Casework

Problem

Let $a$ and $b$ be real numbers, and let $f(x)=|a^2x^2-1|+ax$ . Suppose that

f(x)\geqslant |x|\quad\text{for all }x\in[b,+\infty).

Find the minimum value of

m=a^2b^2+\left(b-\frac 12\right)^2.

Answer

Solution

Difficulty	8/10
Topics	functions, Substitution, Absolute Value, inequality, Casework

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