Final answer:
The value of the directional derivative at the point (1, 1) in the direction (1, 1) is -8/3.
Step-by-step explanation:
The directional derivative of a function at a point in a given direction can be found using the dot product of the gradient vector of the function at that point and the unit vector representing the direction. In this case, the direction is given as (1, 1).
The gradient of the function f(x, y) = 6 - (x²/3) - y² is (∂f/∂x, ∂f/∂y) = (-2x/3, -2y). Evaluating this gradient at the point (1, 1), we get (-2/3, -2).
The unit vector in the direction (1, 1) is found by dividing the direction vector by its magnitude, which gives (1/√2, 1/√2).
Taking the dot product of the gradient vector and the unit vector, we have (-2/3 * 1/√2) + (-2 * 1/√2) = -2/3√2 - 2/√2 = -2(√2 + 3√2)/3√2 = -2(4√2)/3√2 = -8/3.
Therefore, the value of the directional derivative at the point (1, 1) in the direction (1, 1) is -8/3.