Final answer:
The steepest ascent of the function F=4y√(x) at the point (4,1) is given by the gradient vector. Computing the partial derivatives and evaluating them at (4,1), we find the gradient vector to be (1, 8), indicating that the steepest ascent has both x and y components.
Step-by-step explanation:
The problem is asking to find the steepest ascent of the function F = 4y√(x) at the point (4,1). The steepest ascent refers to the direction of the greatest increase in the value of the function, which is given by the gradient vector or the vector of partial derivatives.
To find this, we first need to calculate the partial derivative of F with respect to x and the partial derivative of F with respect to y:
- ∂F/∂x = 4y * (1/2)x^(-1/2) = 2y / √(x)
- ∂F/∂y = 4√(x)
Evaluating these derivatives at the point (4,1):
- ∂F/∂x at (4,1) = 2 * 1 / √(4) = 1
- ∂F/∂y at (4,1) = 4 * √(4) = 8
Therefore, the gradient vector at (4,1) is (1, 8), which means the steepest ascent occurs in the direction of this gradient vector rather than strictly in the x-direction or y-direction. Thus, the correct answer is none of the options provided, as the steepest ascent is in the direction of the gradient vector, which has both x and y components.