help with 57 please Consider the following functions f, points P, and unit vecto

Question

help with 57 please

Consider the following functions f, points P, and unit vectors u. 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. 55. f(x,y,z) =x^2 + 2y^2 + 4z^2 + 10; P(1,0,4); 1/2,0,1/2 56. f(x,y,z)= 4 - x^2 + 3y^2 + z^2/2, P(0,2,-1); 0,1/2, -1/2 57. f(x,y,z) = 1 + 4xyz; P(1, -1, -1); 1/3,1/3, -1/3 58. f(x,y,z) = xy + yz + xz + 4; P(2, -2, 1); 0, -1/2,-1/2 59. f(x,y,z) = 1 + sin (x + 2y - z); P(pi/6,pi/6,-pi/6); 1/3,2/3,2/3

Explanation / Answer

del F = <fx, fy,fz>

fx = 4yz , fy = 4xz , fz = 4xy

at(1,-1,-1) we have fx = 4 , fy = -4 , fz = -4

so del F = <4,-4,-4>

directional derivative= <4,-4,-4>.<1/sqrt(3) , 1/sqrt(3) , -1/sqrt(3)>

= 4 /sqrt(3) -4/sqrt(3)+4/sqrt(3)

= 4/sqrt(3) ------ANSWER

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