Final answer:
The maximum rate of change of the function f(x,y) = 4xy at the point (1,4) is the magnitude of the gradient vector at that point, which is √272, and it occurs in the direction of the gradient vector, which as a unit vector is (16, 4)/√272.
Step-by-step explanation:
To find the maximum rate of change of the function f(x,y) = 4xy at the point (1,4), you need to calculate the gradient of the function and evaluate it at that point. The gradient is a vector that points in the direction of the greatest rate of increase of the function.
The gradient is given by ∇f(x,y) = ∇(4xy) = (4y, 4x). At the point (1,4), ∇f(1,4) = (4*4, 4*1) = (16, 4). The magnitude of this gradient vector is the maximum rate of change, which is √(16^2 + 4^2) = √(256 + 16) = √272.
The direction of this maximum rate of change is the same as the direction of the gradient vector, which can be expressed as a unit vector by dividing the gradient by its magnitude. Thus, the direction is (16, 4)/√272.