lean_workbook_plus_8810

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by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Prove that $\sqrt{x\cos x} \le \frac{x + \cos x}{2}$ for $x\in [0,\frac{\pi}{2}]$ using the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_8810 (x : ℝ) (hx : 0 ≤ x ∧ x ≤ π/2) :
  Real.sqrt (x * Real.cos x) ≤ (x + Real.cos x) / 2   :=  by sorry

Source