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Question Number 2548 by Filup last updated on 22/Nov/15
For a function y=f(x), inflection points/stationary points are when (df/dx)=0. For a function z=f(x, y), can you find these points through a similar method? Is it something like (∂f/∂x)=0 and (∂f/∂y)=0?
Forafunctiony=f(x),inflectionpoints/stationarypointsarewhendfdx=0.Forafunctionz=f(x,y),canyoufindthesepointsthroughasimilarmethod?Isitsomethinglike∂f∂x=0and∂f∂y=0?
Answered by Yozzi last updated on 22/Nov/15
Let f be a function of two variables whose first and second partial derivatives are continous on some open disc B. Suppose further that at the point (a,b) f_x =(∂f/∂x)=0 and f_y =(∂f/∂y)=0. Let p=(∂^2 f/∂x^2 )=f_(xx) , q=(∂^2 f/∂y^2 )=f_(yy) and r=(∂^2 f/(∂y∂x))=f_(xy) (a,b) is a local minimum if pq−r^2 >0 and p>0 (or q>0), (a,b) is a local maximum if pq−r^2 >0 and p<0 (or q<0) (a,b) is a saddle point if pq−r^2 <0 pq−r^2 =0 is inconclusive.
LetfbeafunctionoftwovariableswhosefirstandsecondpartialderivativesarecontinousonsomeopendiscB.Supposefurtherthatatthepoint(a,b)fx=∂f∂x=0andfy=∂f∂y=0.Letp=∂2f∂x2=fxx,q=∂2f∂y2=fyyandr=∂2f∂y∂x=fxy(a,b)isalocalminimumifpq−r2>0andp>0(orq>0),(a,b)isalocalmaximumifpq−r2>0andp<0(orq<0)(a,b)isasaddlepointifpq−r2<0pq−r2=0isinconclusive.
Commented by Filup last updated on 22/Nov/15
My question is what is an open disc and what is f_(xy) ? is (∂^2 f/(∂x∂y)) differentiation with two variables?
Myquestioniswhatisanopendiscandwhatisfxy?is∂2f∂x∂ydifferentiationwithtwovariables?
Commented by Yozzi last updated on 22/Nov/15
f_x is partial differentiation of f w.r.t x f_(xy) =(∂/∂y)(f_x )=(∂/∂y)((∂f/∂x))=(∂^2 f/(∂y∂x)) (mixed 2nd partial derivative) An n−dimensional open disc of radius r is the collection of points less than a distance of r away from a fixed point in Euclidean n− space. (Wolfram Alpha) If n=1 for example, this is an open interval (s,t). So, for n=2 this defines a collection of points in a circle containing the point (a,b) where both f_x and f_(y ) are zero. In this case, once the 1st and 2nd partial derivatives of f exist and the both f_(xy) and f_(yx ) are continuous for the set of points (x,y) in the disc, these two mixed partial 2nd derivatives are equal, i.e f_(xy) =f_(yx) .
fxispartialdifferentiationoffw.r.txfxy=∂∂y(fx)=∂∂y(∂f∂x)=∂2f∂y∂x(mixed2ndpartialderivative)Ann−dimensionalopendiscofradiusristhecollectionofpointslessthanadistanceofrawayfromafixedpointinEuclideann−space.(WolframAlpha)Ifn=1forexample,thisisanopeninterval(s,t).So,forn=2thisdefinesacollectionofpointsinacirclecontainingthepoint(a,b)wherebothfxandfyarezero.Inthiscase,oncethe1stand2ndpartialderivativesoffexistandthebothfxyandfyxarecontinuousforthesetofpoints(x,y)inthedisc,thesetwomixedpartial2ndderivativesareequal,i.efxy=fyx.
Commented by Filup last updated on 22/Nov/15
I see. Very interesting. I must go learn multi−variable calculus!
Isee.Veryinteresting.Imustgolearnmulti−variablecalculus!

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