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Sketch the graph of an arbitrary function \(f\) satisfying the given conditions. State whether the function has any extrema or saddle points. (There are many correct answers.) $$ \begin{aligned} &f_{x}(0,0)=0, f_{y}(0,0)=0\\\ &f_{x}(x, y)\left\\{\begin{array}{ll} <0, & x<0 \\ >0, & x>0 \end{array}, \quad f_{y}(x, y)\left\\{\begin{array}{ll} >0, & y<0 \\ <0, & y>0 \end{array}\right.\right.\\\ &f_{x x}(x, y)>0, f_{y y}(x, y)<0, \text { and } f_{x y}(x, y)=0 \text { for all }(x, y) \end{aligned} $$

Short Answer

Expert verified
The function has a saddle point at (0,0).

Step by step solution

01

Understanding given conditions

The exercise gives conditions for \(f_x\) and \(f_y\), which are the first partial derivatives of \(f\) with respect to \(x\) and \(y\) respectively. At (0,0), \(f_x\) and \(f_y\) are 0, this implies (0,0) is a critical point. For the rest of the values, \(f_x>0\) when \(x>0\) and \(f_x<0\) when \(x<0\), this suggests the function is increasing when \(x\) is increasing and decreasing when \(x\) is decreasing. Moreover, \(f_y>0\) when \(y<0\) and \(f_y<0\) when \(y>0\), which suggests that the function is increasing when \(y\) is decreasing and decreasing when \(y\) is increasing. Additionally, \(f_{xx}>0, f_{yy}<0, \text{and } f_{xy}=0\) for all \((x, y)\), which implies \(f_{xx}(0,0)>0, f_{yy}(0,0)<0, \text{and } f_{xy}(0,0)=0\).
02

Determining extrema or saddle points

The second derivative test for extrema requires to calculate the determinant of the bordered Hessian matrix at the critical point. The determinant \(D=f_{xx}(0,0)f_{yy}(0,0)-(f_{xy}(0,0))^2\) at the critical point is \(>0\) because \(f_{xy}(0,0)=0, f_{xx}(0,0)>0, \text{and } f_{yy}(0,0)<0\). Therefore, \(D<0\). If \(D<0\), the function \(f\) has a saddle point at the critical point. So, there is a saddle point at (0,0).
03

Sketching the graph

From the first partial derivatives conditions, we can infer that the function should be increasing while moving away from the y-axis in the positive x direction, and decreasing while moving towards the y-axis in the negative x direction. Inversely it should be increasing while moving towards the x-axis in the negative y direction, and decreasing while moving away from the x-axis in the positive y direction. Therefore, we would expect a shape that is lowered at the origin, then rises as we move away from the origin along either the positive or negative x-axis and the negative y-axis, but lowers as we move away from the origin along the positive y-axis. Thus, (0,0) acts as a saddle point.

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