ABSTRACT
Given z = f ( x , y) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3706.tif"/> :
determine ∂ z ∂ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3707.tif"/> and ∂ z ∂ y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3708.tif"/>
for stationary points, ∂ z ∂ x = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3709.tif"/> and ∂ z ∂ y = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3710.tif"/> ,
solve the simultaneous equations ∂ z ∂ x = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3711.tif"/> and ∂ z ∂ y = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3712.tif"/> for x and y, which gives the co-ordinates of the stationary points,
determine ∂ 2 z ∂ x 2 , ∂ 2 z ∂ y 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3713.tif"/> and ∂ 2 z ∂ x ∂ y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3714.tif"/>
for each of the co-ordinates of the stationary points, substitute values of x and y into ∂ 2 z ∂ x 2 , ∂ 2 z ∂ y 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3715.tif"/> and ∂ 2 z ∂ x ∂ y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3716.tif"/> and evaluate each,
evaluate ( ∂ 2 z ∂ x ∂ y ) 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3717.tif"/> for each stationary point,
309substitute the values of ∂ 2 z ∂ x 2 , ∂ 2 z ∂ y 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3718.tif"/> and ∂ 2 z ∂ x ∂ y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3719.tif"/> into the equation Δ = ( ∂ 2 z ∂ x ∂ y ) 2 − ( ∂ 2 z ∂ x 2 ) ( ∂ 2 z ∂ y 2 ) and evaluate, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3720.tif"/>
if Δ > 0 then the stationary point is a saddle point
if Δ < 0 and ∂ 2 z ∂ x 2 < 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3721.tif"/> , then the stationary point is a maximum point, and
if Δ < 0 and ∂ 2 z ∂ x 2 < 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3722.tif"/> , then the stationary point is a minimum point
