Direction and Rate of Change for Function at a Point

For the function $f(x, y) = x^2y - xy^2$ at the point (1, -1), find the direction and rate of greatest increase, greatest decrease, and a direction of no change.

This problem involves understanding how a multivariable function behaves at a given point and finding various directional derivative properties. When dealing with functions of two variables, such as $f(x, y) = x^2y - xy^2$, at a specific point, it is crucial to understand the role of gradients. The gradient vector is a fundamental concept in multivariable calculus as it points in the direction of the steepest ascent of the function and its magnitude describes how fast the function is increasing. Hence, finding the gradient is the first step in addressing this problem, as it directly gives the direction of greatest increase.

The rate of greatest increase is the magnitude of the gradient vector itself. To find the direction of greatest decrease, one would consider the negative of the gradient vector, effectively pointing in the opposite direction. For the direction of no change, we look for a direction perpendicular to the gradient, meaning any vector orthogonal to it would result in zero change along that direction. This concept reinforces the orthogonal nature of gradients and level curves or surfaces, where each level curve is tangent to the directions of no change.

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