   Chapter 14, Problem 48RE

Chapter
Section
Textbook Problem

Find the direction in which f(x, y, z) = zexyincreases most rapidly at the point (0, 1, 2). What is the maximum rate of increase?

To determine

To find: The maximum rate of change of f(x,y,z)=zexy at the point (0,1,2) 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 function is, f(x,y,z)=zexy .

Calculation:

The maximum rate of change of f(x,y,z)=zexy at the point (0,1,2) is computed as follows.

f(x,y,z)=fx,fy,fz=x(zexy),y(zexy),z(zexy)=(yzexy),(xzexy)(exy)

Thus, f(x,y,z)=(yzexy),(xzexy)(exy) .

Substitute x = 0, y = 1, z = 2 in f(x,y) and find the value of f(0,1,2)

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