Question

If the directional derivative of $= ax^2y+by²z+cz^2x at (1, 1, 1) has maximum magnitude 15 in the direction parallel to the line x-1/2=y-3/-2=z
find a, b and c.​

Answer 1

Explained in attachment

Answer 2

Answer:

Please go through the solution below:

Explanation:

We have

∇f=(ay2+3cx2z2)i^+(2axy+bz)j^+(by+2cx3z)k^

so that

∇f(1,2,−1)=(4a+3c)i^+(4a−b)j^+(2b−2c)k^.

The directional derivative of f in a direction parallel to z-axis, that is, k^ is

∇f(1,2,−1)⋅k^=(4a+3c)⋅0+(4a−b)⋅0+(2b−2c)⋅1

Since this directional derivative has a magnitude of a maximum of 64, we must have

4a+3c=0,         4a−b=0       and        2b−2c=64.

Solving these, we get

a=6,

b=24

and c= −8.