Lokale Extrempunkte
f(x, y) = 4·x^2 - 3·x·y
f'(x, y) = [8·x - 3·y, - 3·x] = [0, 0]
f''(x, y) = [8, -3; -3, 0] --> Sattelpunkt im Ursprung
Randextrema über Lagrange
L(x, y, k) = 4·x^2 - 3·x·y - k·(x^2 + y^2 - 1)
L'x(x, y, k) = 8·x - 3·y - 2·k·x = 0 --> k = 4 - 3·y/(2·x)
L'y(x, y, k) = -3·x - 2·k·y = 0 --> k = - 3·x/(2·y)
Die Werte für k gleichsetzen
4 - 3·y/(2·x) = - 3·x/(2·y)
8·x·y - 3·y^2 = - 3·x^2 --> y = -x/3 ∨ y = 3·x
Das in die Nebenbedingung einsetzen
x^2 + (-x/3)^2 = 1
x = -3·√10/10 --> y = -(-3·√10/10)/3 = √10/10 --> HP(-0.9487 | 0.3162 | 4.5)
x = 3·√10/10 --> y = -(3·√10/10)/3 = -√10/10 --> HP(0.9487 | -0.3162 | 4.5)
x^2 + (3·x)^2 = 1
x = -√10/10 --> y = 3·(-√10/10) = -3·√10/10 --> TP(-0.3162 | -0.9487 | -0.5)
x = √10/10 --> y = 3·(√10/10) = 3·√10/10 --> TP(0.3162 | 0.9487 | -0.5)