Final answer:
Therefore, the correct answer is (a) Upward and to the right.The gradient vector of a function points in the direction of the steepest increase in the function.
Step-by-step explanation:
The question asks to sketch the gradient vector of the function f at the point (4,6). To answer this, we must consider the behavior of the level curves of the function at that point. Gradient vectors are always perpendicular to level curves and point in the direction of the steepest ascent on the graph of the function.
Since detailed information about the direction of the level curves at the point (4,6) is not provided, we cannot accurately sketch the vector. Generally, if the level curves are inclined such that they increase in value as we move to the right and upwards, the gradient vector will point in that direction, which is option a) Upward and to the right. Conversely, if level curves increase as we move left and downwards, the gradient would be in that direction, which corresponds to option b) Downward and to the left. The other options can be inferred similarly based on the slope of the level curves.
To determine the direction of the gradient vector at a specific point, we can look at the level curves of the function. If the level curves are upward and to the right, then the gradient vector at that point will also be upward and to the right.
The gradient vector at a point is perpendicular to the level curve and points in the direction of steepest ascent. Without specific details of the level curves at the point (4,6), we cannot accurately sketch the gradient vector.