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Determine the direction in which the function is increasing most rapidly at the point P_0. f(x,...

Question:

Determine the direction in which the function is increasing most rapidly at the point {eq}P_0 {/eq}.

{eq}f(x, y) = xy^2 - yx^2 \;, P_0=(-1,-,1) {/eq}

Gradient of a Function:

Given a function of two variables, the direction of maximum increase at a point P(x,y) is defined by

the gradient of the function, i.e.

{eq}\bigtriangledown f (x,y) = f_x \mathbf{x} + f_y \mathbf{y} {/eq}

where {eq}f_{x,y} {/eq} are the partial derivatives of the function.

Answer and Explanation:

The gradient of the function

{eq}f(x, y) = xy^2 - yx^2 {/eq}

at point (-1, -1) given by

{eq}\bigtriangledown f (x,y) = f_x \mathbf{x} + f_y...

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Directional Derivatives, Gradient of f and the Min-Max

from GRE Math: Study Guide & Test Prep

Chapter 14 / Lesson 6
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