Problem
ALG-B2-M05-P006 Tangent to the reciprocal
#6
★★★☆☆ Level 3 of 5
Prove for \(x>0\): \[\frac1x\ge2-x.\]
Hint. Multiply by \(x\) and obtain a square.
Since \(x>0\), the inequality is equivalent to \(1\ge2x-x^2\), or \((x-1)^2\ge0\). This is the tangent to \(1/x\) at \(1\).
Useful for later problems where each fraction is replaced by a linear bound.