Consider the function f (x comma y comma z )equals 1 plus 4 xyz, the point p(negative 1 comma negative 1 comma 1 ), and the unit vector bold uequalsleft angle startfraction 1 over startroot 3 endroot endfraction comma startfraction 1 over startroot 3 endroot endfraction comma startfraction 1 over startroot 3 endroot endfraction right angle . a. compute the gradient of f and evaluate it at p. b. find the unit vector in the direction of maximum increase of f at p. c. find the rate of change of the function in the direction of maximum increase at p. d. find the directional derivative at p in the direction of the given vector.

This is pretty hard to decipher.

Consider the function f (x comma y comma z )equals 1 plus 4 xyz,

the point P(negative 1 comma negative 1 comma 1 ),

P(-1, -1, 1)

and the unit vector Bold uequalsleft angle StartFraction 1 Over StartRoot 3 EndRoot EndFraction comma StartFraction 1 Over StartRoot 3 EndRoot EndFraction comma StartFraction 1 Over StartRoot 3 EndRoot EndFraction right angle .

Huh?  Oh, I see.  We won't stress about the vector notation.

a. Compute the gradient of f and evaluate it at P.

P(-1, -1, 1)

b. Find the unit vector in the direction of maximum increase of f at P.

That's just the normalized version of the above.  It has a magnitude

so we have to divide 4 by that, so 1/√3.

The unit direction vector is

Sometimes they seek the teacher friendly

c. Find the rate of change of the function in the direction of maximum increase at P.

We already did, that's the magnitude of the gradient,

d. Find the directional derivative at P in the direction of the given vector.

We want the total derivative in the direction of u, so that's the weighted average of the partials.  For unit direction u=(a,b,c) we can write it

For us that's

i think c. but don't trust it.

step-by-step explanation:

i think the answer is b