ABSTRACT
Given https://www.w3.org/1998/Math/MathML"> z = f ( x , y) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3706.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> :
determine https://www.w3.org/1998/Math/MathML"> ∂ z ∂ x https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3707.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ∂ z ∂ y https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3708.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
for stationary points, https://www.w3.org/1998/Math/MathML"> ∂ z ∂ x = 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3709.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ∂ z ∂ y = 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3710.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ,
solve the simultaneous equations https://www.w3.org/1998/Math/MathML"> ∂ z ∂ x = 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3711.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ∂ z ∂ y = 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3712.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for x and y, which gives the co-ordinates of the stationary points,
determine https://www.w3.org/1998/Math/MathML"> ∂ 2 z ∂ x 2 , ∂ 2 z ∂ y 2 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3713.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ∂ 2 z ∂ x ∂ y https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3714.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
for each of the co-ordinates of the stationary points, substitute values of x and y into https://www.w3.org/1998/Math/MathML"> ∂ 2 z ∂ x 2 , ∂ 2 z ∂ y 2 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3715.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ∂ 2 z ∂ x ∂ y https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3716.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and evaluate each,
evaluate https://www.w3.org/1998/Math/MathML"> ( ∂ 2 z ∂ x ∂ y ) 2 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3717.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for each stationary point,
substitute the values of https://www.w3.org/1998/Math/MathML"> ∂ 2 z ∂ x 2 , ∂ 2 z ∂ y 2 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3718.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ∂ 2 z ∂ x ∂ y https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3719.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> into the equation https://www.w3.org/1998/Math/MathML"> 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-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3720.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
if Δ > 0 then the stationary point is a saddle point
if Δ < 0 and https://www.w3.org/1998/Math/MathML"> ∂ 2 z ∂ x 2 < 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3721.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , then the stationary point is a maximum point, and
if Δ < 0 and https://www.w3.org/1998/Math/MathML"> ∂ 2 z ∂ x 2 < 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429294402/7a019bd2-8e3e-42f5-a1b0-42593cc08f27/content/eq3722.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , then the stationary point is a minimum point
