EXAMPLE 3: Find the local maximum and minimum values and saddle points of f(x, y) = x^4 + y^4 - 4xy + 1.
SOLUTION: We first locate the critical points.
fx = 4x^3 - 4y
fy = 4y^3 - 4x
Setting these partial derivatives equal to 0, we obtain the equations:
x^8 - x = 0
y^3 - x = 0
To solve these equations, we substitute y = x^3 from the first equation into the second one. This gives:
0 = x^9 - x = x(x^8 - 1) = x(x^4 - 1)(x^4 + 1) = x(x^2 - 1)(x^2 + 1)(x^4 + 1)
so there are three real roots: x = 0, 1, -1.
The three critical points are (0, 0), (1, 1), (-1, -1). Next we calculate the second partial derivatives and D(x, y).
fxx = 12x^2 - 0
fxy = -4
fyy = 12y^2 - 0
D(x, y) = fxxfyy - (fxy)^2 = 144x^2y^2 - 16
Since D(0, 0) = -16 < 0, it follows from the Second Derivative Test that the origin is a saddle point; that is, f has no local maximum or minimum at (0, 0). Since D(1, 1) = 128 > 0 and fxx(1, 1) = 12 > 0, we see that f(1, 1) = -1 is a local minimum. Similarly, we have D(-1, -1) = 128 > 0 and fxx(-1, -1) = 12 > 0, so f(-1, -1) = -1 is also a local minimum.