Question

A critical point of f(x, y) is any point where both partial derivatives vanish. Find all critical points of (a) f(x, y) = x 2 − 3xy + y 2 + 2x + 5 (b) f(x, y) = x 2 − 2xy + y 2 (c) f(x, y) = Ax2 + Bxy + Cy2 + Dx + Ey + F

Answer 1

Answer:

Step-by-step explanation:

$(a) f(x, y) = x^ 2-3xy + y ^2 + 2x + 5$

Let us find out partial derivatives

$f_x =2x-3y+2\\f_y = -3x+2y$

Equate these to 0 and solve

we get

$6x-9y+6 =0\\-6x+4y=0\\-5y+6 =0\\y=1.2 \\x=0.8$

Critical point (0.8,1.2)

b) $f(x, y) = x^ 2 - 2xy + y^ 2\\f_x=2x-2y=0\\f_y = -2x+2y =0\\x=y$

So (x=y) is the solution

c) $f(x, y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F\\f_x=2Ax+By+D\\f_y=Bx+2cY+E$

Equate to 0 and solve

2ABx + B^2y +DB =0

2ABx+4ACy +2AE =0

$y =\frac{BD-2AE}{B^2-4AC}$

Similarly for x

4ACX + 2BCy +2CD =0

B^2 x +2BCy+BE=0

$x=\frac{BE-2CD}{B^2-4AC}$