Consider the function fx y z 3x2y2 2yz What is the maximum

Consider the function f(x, y, z) = 3x^2y^2 + 2yz What is the maximum rate of change of f(x, y, z) at (-1, 2, 3)?. In what direction does this occur? Find the directional derivative of f(x, y, z) at (-1, 2, 3) in the direction of i with rightarrow - k with rightarrow

Solution

a) Given that f (x,y,z) = 3x²y² + 2yz

        (x,y,z) = (-1,2,3)

Maximum rate of change = | f (x,y,z)|

f = </x, /y, /z>

    = < 6xy², 6x²y+2z, 2y >

f (-1,2,3) = < 6.-1.2², 6.(-1)².2 + 2.3 ,2.2 >

                 = < -24, 18, 4 >

f (-1,2,3) = < -24, 18, 4 >

|f (-1,2,3)| = [ (-24)² + (18)² + (4)²]1/2 = (916)^1/2

Therefore,

Maximum rate of change = (916)^1/2

b)    Directional derivative =   f (-1,2,3) . u

Given that f (x,y,z) = 3x²y² + 2yz

        (x,y,z) = (-1,2,3)

Given that v = i-k

   u = v/|v| = (1/sqrt 2) (i-k) = (1/sqrt 2) <1,-1>

From part (a), f (-1,2,3) = < -24, 18, 4 >

Hence,

Directional derivative =   f (-1,2,3) . u

                              = < -24, 18, 4 >. (1/sqrt 2) <1,-1>

                             = (1/sqrt 2) (-24 -4)

                            = -28/ 2^1/2