To find the direction for which the directional derivative is a maximum at the point (2, 1), we need to calculate the gradient of the function at that point and then find the unit vector in that direction.
The gradient of the function f(x, y) = 5x + 7y^2 is given by:
∇f(x, y) = (∂f/∂x, ∂f/∂y) = (5, 14y)
At the point (2, 1), the gradient is ∇f(2, 1) = (5, 14).
To find the unit vector in the direction of the gradient, we normalize the gradient vector:
Unit vector = (∇f(2, 1)) / ||∇f(2, 1)||
where ||∇f(2, 1)|| is the magnitude of the gradient vector, which is √(5^2 + 14^2).
Calculate the unit vector and you'll have the direction for which the directional derivative is a maximum at the point (2, 1).
The direction for which the directional derivative of the function is a maximum at the point (2, 1) is approximately (0.347, 0.938).