Question

Find the maximum rate of change of f at the given point and the direction in which it occurs. The function f(x, y) = 4yx. Given the point (4, 9), what is the maximum rate of change? What is the direction vector?

Answer 1

The maximum rate of change of f(x, y) = 4yx at the point (4, 9) is √1552, and the direction vector in which this occurs is (36, 16)/√1552.

Step-by-step explanation:

The maximum rate of change of a function at a given point is the magnitude of the gradient vector at that point. To find the maximum rate of change of f(x, y) = 4yx at the point (4, 9), we calculate the partial derivatives of f with respect to x and y, which are f_x = 4y and f_y = 4x, respectively.

At the point (4, 9), the partial derivatives are f_x(4,9) = 4*9 = 36 and f_y(4,9) = 4*4 = 16. The gradient vector ∇f at (4, 9) is then (36, 16), which gives the direction of the maximum rate of change. The magnitude of this vector, ||∇f||, is √(36^2 + 16^2) = √(1296 + 256) = √1552, which is the maximum rate of change at the point (4, 9).

The direction vector indicating where this maximum rate of change occurs is the unit vector in the direction of the gradient, which is ∇f / ||∇f|| = (36, 16)/√1552.