   Chapter 14.6, Problem 25E

Chapter
Section
Textbook Problem

Find the maximum rate of change of f at the given point and the direction in which it occurs.25. f(x, y, z) = x/(y + z), (8, 1, 3)

To determine

To find: The maximum rate of change of f(x,y,z)=xy+z at the point (8,1,3) and the direction in which the maximum rate of change occurs.

Explanation

Theorem used:

“Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative Duf(x) is |f(x)| and it occurs when u has the same direction as the gradient vector f(x)

Given:

The given functionis, f(x,y,z)=xy+z .

Calculation:

The maximum rate of change of f(x,y,z)=xy+z at the point (8,1,3) is computed as follows,

f(x,y,z)=fx,fy,fz=x(xy+z),y(xy+z),z(xy+z)=(1y+z),[x(1(y+z)2(1))],[x(1(y+z)2(1))]=(1y+z),(x(y+z)2),(x(y+z)2)

Thus, f(x,y,z)=(1y+z),(x(y+z)2),(x(y+z)2)

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