Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f(x,y)=x3−6xy+8y3

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Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.  f(x,y)=x3−6xy+8y3

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Jeffrey Jordon · Accepted Answer

Find the critical points using the given function, and identify whether they are the local minimum, maximum, or saddle point.f(x,y)=x3−6xy+8y3Identify first derivatives in terms of&amp;nbsp;Fx(x,y)&amp;nbsp;and&amp;nbsp;Fy(x,y)Fx(x,y)=3x2−6yFy(x,y)=−6x+24y2Find the critical points of x by solving for y in&amp;nbsp;Fx&amp;nbsp;and substituting result into&amp;nbsp;FyFx(x,y)=3x2−6y→3x2=6y→12x2=yFy(x,y)=−6x+24(12x2)2→−6x+6x4=6x(x3−1)=0→x=0&amp;nbsp;and&amp;nbsp;x=1Find the critical values of y associated with the critical values found for x by substituting into&amp;nbsp;FxFx(0,y)=3x2−6y→0=6y→y=0Fx(1,y)=3x2−6y→3=6y→y=12using&amp;nbsp;Fx(x,y)&amp;nbsp;and&amp;nbsp;Fy(x,y)&amp;nbsp;find the second derivatives in terms of&amp;nbsp;Fxx(x,y),&amp;nbsp;Fyy(x,y)&amp;nbsp;and&amp;nbsp;Fxy(x,y)Fxx(x,y)=6xFyy(x,y)=48yFxy=−6When the rules are defined, use the second derivative test.D=D(x,y)=Fxx(x,y)⋅Fyy(x,y)−(Fxyx,y)2D&amp;gt;0&amp;nbsp;and&amp;nbsp;Fxx(x,y)&amp;lt;0, then&amp;nbsp;F(x,y)&amp;nbsp;is local maximum&amp;nbsp;D&amp;lt;0&amp;nbsp;then&amp;nbsp;F(x,y)&amp;nbsp;is a saddle pointThe critical points identified as (0,0) and&amp;nbsp;(1,12)&amp;nbsp;are to be used to solveD=D(0,0)=0⋅0−(−6)2=36;&amp;nbsp;D=D(1,12)=6⋅24−(−6)2=108Compare D values of&amp;nbsp;Fxx(x,y)&amp;nbsp;to indentify critical points as local maximum, minimum or saddle pointsD=D(0,0)=−36→D&amp;lt;0&amp;nbsp;Saddle pointD=D(1,12)=108,&amp;nbsp;Fxx(1,12)&amp;gt;0D(1,12)&amp;nbsp;is local minimum

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f(x,y)=x^3-6xy+8y^3

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