Please help with #21. Thanks On this page, Exercises 19–24 focus on identifying the directions in which specified functions increase and decrease most rapidly at given points, \( P_0 \). The objective is also to find the derivatives of the functions in these specified directions.19. \( f(x, y) = x^2 + xy + y^2, \quad P_0(-1, 1) \)20. \( f(x, y) = x^2 y + e^{xy} \sin y, \quad P_0(1, 0) \)21. \( f(x, y, z) = \frac{x}{y} - yz, \quad P_0(4, 1, 1) \)22. \( g(x, y, z) = xe^{y} + z^2, \quad P_0(1, \ln 2, \frac{1}{2}) \)23. \( f(x, y, z) = \ln (xy) + \ln (yz) + \ln (xz), \quad P_0(1, 1, 1) \)24. \( h(x, y, z) = \ln (x^2 + y^2 - 1) + y + 6z, \quad P_0(1, 1, 0) \)For each exercise, you must first determine the gradient of the function, which points in the direction of the steepest ascent. The negative gradient then gives the direction of the steepest descent. Calculating the derivatives of the functions in these directions involves finding the directional derivative along the gradient and its opposite.

Name: Please help with #21. Thanks On this page, Exercises 19–24 focus on id
Uploaded: 2024-03-20T07:07:37.000Z
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Description: Please help with #21. Thanks On this page, Exercises 19–24 focus on identifying the directions in which specified functions increase and decrease most rapidly at given points, \( P_0 \). The objective is also to find the derivatives of the functions