Question

Consider the function f(x,y)=xy+xz+yz+4 at point p=(2,-1,1)
a)find the unit vector in direction of p.
b) find the directional derivative at p in the direction of <0,-1/(sqrt2),-1/(sqrt2)>

Answer 1

|<b>p</b>| = √(4+1+1) = √6
<b>u</b>= <b>p</b>/|<b>p</b>| = (2/√6,-1/√6,1/√6)

Now, with <b>u</b> = <0,-1/√2,-1/√2)>
∇<sub<<b>u</b></sub>f = ∇f•<b>u</b>
= (y+z)(0) + (x+z)(-1/√2) + (x+y)(-1/√2)
= (0)(0) - 1/√2 (2+1) - 1/√2 (2-1)
= -4/√2 = -2√2

Answer 2

Oh crap typo error, should read find the unit vector in direction of maximum increase of f in the direction of P.