Partial Differentiation 2

f(x,y,z)=axy^2+byz+cz^2 x^3
grad(f)=< ay^2+3cz^2 x^2, 2axy+bz, by+2cz x^3>
For P(1, 2, -1), grad(f)=<4a+3c, 4a-b, 2b-2c>
(i) grad(f)=<0,0,∂f/∂z>, 則 4a+3c=0, 4a-b=0, 2b-2c>=0
a: b: c= 1: 4: -4/3, 得(a, b, c)=(t, 4t, -4t/3), t>=0
(ii) v=<1, 2, 3>, |v|=√14, unit direction=<1, 2, 3>/√14
directional derivative=grad(f) dot <1,2,3>/√14
=6(b-c)/√14=32t/√14 (t>=0)

註:max{ directional derivative }=| grad(f) | ,
(i) |grad(f)| = ∂f/∂z, 則 (∂f/∂x)^2+(∂f/∂y)^2+(∂f/∂z)^2= (∂f/∂z)^2, 且(∂f/∂z)>= 0
故(∂f/∂x)=(∂f/∂y)=0, or grad(f)=<0,0, ∂f/∂z>且(∂f/∂z)>0