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Mathematics-Online problems:

Interactive Problem 413: Critical Points and Sketch of a function of two Variables

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Given the function

$\displaystyle f(x,y)=(1-x^2-y)(1-x^2+y)x=x^5-2x^3-xy^2+x \, . $

Sketch the curve with $ f(x,y)=0$ in the $ xy$-plane and indicate where $ f(x,y) >0$ (mark with +) or $ f(x,y)<0$ (mark with -) holds.

 keine Angabe Sketch 1 Sketch 2
   \includegraphics[width=0.5\linewidth]{bild1} \includegraphics[width=0.5\linewidth]{bild2}
   Sketch 3 Sketch 4
   \includegraphics[width=0.5\linewidth]{bild3} \includegraphics[width=0.5\linewidth]{bild4}

Find $ f_x$ and $ f_y$.

$ f_x\ =\ $$ x^4+$$ x^2$+$ xy+$$ y^2$+

$ f_y\ =\ $$ x^4+$$ x^2$+$ xy+$$ y^2$+

Insert in the following table all critical points of $ f$ and mark their typ. (Leave unused columns empty - does not work in the online version!)

point              
local minimum              
lokal maximum              
saddle point              

Give the critical points in an ascending order. (Start with the point whose $ x$-value is minimal. If two $ x$-values are equal then give at first the point with smaller $ y$-value.)

$ x_1=$, $ y_1=$    
n.a.
local maximum
local minimum
saddle point
$ x_2=$ $ \Big/\sqrt{\vphantom{\frac{1}{1}}}$, $ y_2=$    
n.a.
local maximum
local minimum
saddle point
$ x_3=$, $ y_3=$    
n.a.
local maximum
local minimum
saddle point
$ x_4=$, $ y_4=$    
n.a.
local maximum
local minimum
saddle point
$ x_5=$ $ \Big/\sqrt{\vphantom{\frac{1}{1}}}$, $ y_5=$    
n.a.
local maximum
local minimum
saddle point
$ x_6=$, $ y_6=$    
n.a.
local maximum
local minimum
saddle point


   

(Source unknown)

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  automatically generated: 8/11/2017