Answer to Question #104344 in Calculus for Emmanuel

Question #104344
Please how do we distinguish or know when it is a minimum or maximum or saddle point in an extrema of two or more variables(multivariable) function.
1
Expert's answer
2020-03-02T13:58:22-0500

To  distinguish between minimum or maximum or saddle points one can use the second partial derivative test.


Let's define the  Hessian matrix:

H(x,y)=(fxx(x,y)fxy(x,y)fyx(x,y)fyy(x,y))H(x,y) = \begin{pmatrix}f_{xx}(x,y) &f_{xy}(x,y)\\f_{yx}(x,y) &f_{yy}(x,y)\end{pmatrix} ,

with partial derivatives of the function f(x,y)f(x,y) .


Let D(x,y)D(x,y) be the determinate of H(x,y)H(x,y) :

D(x,y)=fxx(x,y)fyy(x,y)(fxy(x,y))2D(x,y)=f_{xx}(x,y)f_{yy}(x,y) - \left( f_{xy}(x,y) \right)^2 .


If x=a,y=bx = a, y = b - is critical point of function, then we have the following distinguish rule:

  1. If D(a,b)>0D(a,b)>0 and fxx(a,b)>0f_{xx}(a,b)>0 then (a,b)(a,b) is a local minimum of f(x,y)f(x,y) .
  2. If D(a,b)>0D(a,b)>0 and fxx(a,b)<0f_{xx}(a,b)<0 then (a,b)(a,b) is a local maximum of f(x,y)f(x,y) .
  3. If D(a,b)<0D(a,b)<0 then (a,b)(a,b) is a saddle point of f(x,y)f(x,y) .
  4. If D(a,b)=0D(a,b)=0 then the second derivative test is inconclusive, and one should use Higher-order derivative test in order to classify the critical point.

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Comments

Assignment Expert
02.03.20, 21:38

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Emmanuel
02.03.20, 21:09

Thanks

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