Solve Schaum's Vector Analysis Prob 4.65: Find Constants a,b,c

[likephysics]

Homework Statement

Schaum's vector analysis prob 4.65
Find the values of constants a,b,c so that directional derivative of phi=axy^2+byz+cz^2x^3 at (1,2,-1) has a maximum of magnitude 64 in a direction parallel to the z axis.

Homework Equations

The Attempt at a Solution

Directional derivative is maximum along grad(phi). After taking grad(phi) at (1,2,-1) I got
(4a+3c)i+(4a-b)j+(2b-2c)k
I don't know how to make use of the 'direction parallel to z axis' condition.

 

[rl.bhat]

If grad(phi) is maximum in Z-axis, it may be zero in x and y-axis. Then
(2b - 2c) = 64.
(4a+3c) = (4a-b) = 0.
Solve these equations to find a, b and c.

 

[likephysics]

Got it! Thanks.