   Chapter 14, Problem 2RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.There exists a function f with continuous second-order partial derivatives such that fx(x, y) = x + y2 and fy(x, y) = x − y2.

To determine

Whether the statement, “There exist a function f with continuous second ordered partial derivatives such that fx(x,y)=x+y2andfy(x,y)=xy2 ” is true or false.

Explanation

Theorem used: Clairaut’s Theorem.

“Suppose f is defined on a disk D that contains the point (a,b) . If the functions fxy and fyx are both continuous on D, then fxy(a,b)=fyx(a,b) ”.

Reason:

The given statement is false by Clairaut’s Theorem.

Find the value of fxy(x,y) ,

fxy(x,y)=y(fx)=y(x+y2)=0+2y=2y

Thus, the value of fxy(x,y) is 2y

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