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) = 3y²x, is evaluated at the point (4, 8). What is the maximum rate of change and the direction vector?

Answer 1

The maximum rate of change of the function f(x, y) = 3y²x at the point (4, 8) is √(192² + 192²), and it occurs in the direction of the gradient vector, which is normalized to <1/√2, 1/√2>.

Step-by-step explanation:

The maximum rate of change of a function at a given point occurs in the direction of the gradient of the function at that point. For the function f(x, y) = 3y²x, we need to find the gradient vector by taking the partial derivatives with respect to x and y. The partial derivative with respect to x (fx) is 3y², and the partial derivative with respect to y (fy) is 6yx. At the point (4, 8), these are fx(4, 8) = 3(8)² and fy(4, 8) = 6(4)(8).

Thus, the gradient vector at (4, 8) is <192, 192>. The magnitude of this vector gives us the maximum rate of change, which is calculated using the Euclidean norm: √(192² + 192²).

The direction vector of the maximum rate of change is the normalized gradient vector, which is the gradient vector divided by its magnitude, resulting in <1/√2, 1/√2>.