Bonsoir
On pose P(x) = x⁴ - 2x³ + 3x² - 2x + 1
P'(x) = 4x³ - 6x² + 6x - 2
P'(1/2) = 4/8 - 6/4 + 6/2 - 2 = 1/2 - 3/2 + 3 - 2 = 0
Ce polynôme admet donc un minimum en 1/2
P(1/2) = 3/5
On a donc P(x/y) ≥ 3/5 ert a fortiori P(x/y) ≥ 0
D'où
(x/y)⁴ - 2 (x/y)³ + 3(x/y)² - 2x/y + 1 ≥ 0
⇔ (x/y)² - 2 x/y + 3 -2 y/x + (y/x)² ≥ 0
⇔ 3 + (x/y)² + (y/x)² ≥ 2x/y + 2y/x
⇔ 3 + x²/y² + y²/x² ≥ 2 (x/y + y/x)
