   Chapter 14, Problem 47RE

Chapter
Section
Textbook Problem

Find the maximum rate of change of f ( x , y ) = x 2 y + y at the point (2, 1). In which direction does it occur?

To determine

To find: The maximum rate of change of f(x,y)=x2y+y at the point (2,1) 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)=x2y+y .

Calculation:

The maximum rate of change of f(x,y)=x2y+y at the point (2,1) is computed as follows.

f(x,y)=fx,fy=x(x2y+y),y(x2y+y)=(2xy),(x2+12y)

Thus, f(x,y)=(2xy),(x2+12y) .

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

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