兩題微積分

1. w^2 + wsin(xyz) = 0

w = -sin(xyz)

wx = -yzcos(xyz)


2. f(x,y) = x^3 + y^3 + 3xy - 2

fx = 3x^2 + 3y, fy = 3y^2 + 3x

fxx = 6x, fyy = 6y, fxy = 3

Put fx = fy = 0

x^2 = -y or y^2 = -x

We get y^2 = x^4

So, x^4 + x = 0

x(x^3 + 1) = 0

x = 0 or -1

For x = 0, y = 0. For x = -1, -1

For (0,0), fxxfyy - (fxy)^2 = (0)(0) - (3)^2 = -9 < 0

For (-1,-1), fxxfyy - (fxy)^2 = (-6)(-6) - (3)^2 = 27 > 0

And fxx = -6 < 0

So, (0,0) is a saddle point

(-1,-1) is a maximum point.