The maximum of a quadratic expression on an ellipse

Proved

WorkbookCorrected.plus_13309

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

corrected-formalizationlean-workbooksource-checked

For real numbers $x,y$ satisfying $4x^2+9y^2=36$ , the maximum of $x^2+\frac23xy+\frac32y^2$ is $10$ . It is attained at $(x,y)=(6\sqrt5/5,2\sqrt5/5)$ .

Formalization Note: The original source asks for an extremum, while its formal statement supplied only a bound. This complete formulation includes both the universal bound and existence of an attaining point.

Source: InternLM Lean-Workbook, record lean_workbook_plus_13309 (Apache-2.0).

Preamble

import Mathlib

Formal statement

theorem WorkbookCorrected.plus_13309 : (∀ x y : ℝ, 4*x^2+9*y^2=36 → x^2+(2/3)*x*y+(3/2)*y^2 ≤ 10) ∧
    (∃ x y : ℝ, 4*x^2+9*y^2=36 ∧ x^2+(2/3)*x*y+(3/2)*y^2=10) := by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_13309; Apache-2.0

View graph