Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises - Page 967: 97


A function $f(x,y)$ does not exist

Work Step by Step

Since, $f_{x}(x, y)=x+4y \\ f_{xy}(x, y)=4$ ...(1) $ f_{y}(x, y)=3x-y \\ f_{yx}(x, y)=3$ ...(2) As per Clairaut's Theorem we should have $f_{xy}(x, y)=f_{yx}(x, y)$ Since, the second order derivatives are not equal in the both equations (1) and (2), thus, both $f_{xy}$ and $f_{yx}$ are continuous, This means that such a function $f(x,y)$ does not exist.
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