Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 13 - Functions of Several Variables - 13.7 Exercises - Page 933: 45

Answer

$$\left( {0,0,0} \right)$$

Work Step by Step

$$\eqalign{ & z = 5xy \cr & 5xy - z = 0 \cr & {\text{Consider}} \cr & F\left( {x,y,z} \right) = 5xy - z \cr & {\text{Calculating the partial derivatives}} \cr & {F_x}\left( {x,y,z} \right) = 5y \cr & {F_y}\left( {x,y,z} \right) = 5x \cr & {F_z}\left( {x,y,z} \right) = - 1 \cr & {\text{Find the gradient }}\nabla F\left( {x,y,z} \right) \cr & \nabla F\left( {x,y,z} \right) = 5y{\bf{i}} + 5x{\bf{j}} - {\bf{k}} \cr & {\text{Find the point}}\left( {\text{s}} \right){\text{ on the surface at which the tangent plane }} \cr & {\text{is horizontal}}. \cr & 5y = 0 \to y = 0 \cr & 5x = 0 \to x = 0 \cr & z = 5xy \cr & z = \left( 0 \right)\left( 0 \right) \cr & z = 0 \cr & {\text{The point is}} \cr & \left( {0,0,0} \right) \cr} $$
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