1. The Function \(f\left( x,y \right)={{\left( x-3 \right)}^{2}}-9{{y}^{3}}+3y+37\)  is unconstrained.  Find all relative extremes if any exist.

1. The function might not have any relative extremes because of its cubic nature. However, we must still look for them by finding \({{f}_{x}}\) and \({{f}_{y}}\) setting both equal to zero, then solving the system.
   What is…
   \({{f}_{x}}\) ?
   \({{f}_{y}}\) ?
   \({{f}_{xx}}\) ?
   \({{f}_{yy}}\) ?
   \({{f}_{xy}}\) ?
2. Solve the System \({{f}_{x}}=0\) and \({{f}_{y}}=0\)
   What are the ordered pairs you found?    ____________________________
3. Suppose D<0. What is the nature of the stationary point?
   Minimum, Maximum, Saddle, or Unknown from this information
4. Suppose D>0 and \({{f}_{xx}}>0\). What is the nature of the stationary point?
   Minimum, Maximum, Saddle, or Unknown from this information
5. Suppose D=0 and \({{f}_{xx}}>0\). What is the nature of the stationary point?
   Minimum, Maximum, Saddle, or Unknown from this information
6. Calculate the determinate for all points you found when solving \({{f}_{x}}=0\) and \({{f}_{y}}=0\).

From this complete the following:

(Use radical where necessary for exact answers) (Enter "none" if the type of stationary point does not appear.)

The relative minimum is at (x,y)= _____________

The relative maximum is at (x,y)=_______________

There is a saddle at (x,y)= ________________

2) The function \[f\left( x,y \right)=-9{{\left( x-5 \right)}^{2}}-5{{\left( y-9 \right)}^{2}}+1395\] is constrained to the region \(R:\,\,0\le x\le 31,\,\,0\le y\le 47\). Find the absolute extrema of the function.

1. Find the relative extrema using the first and second derivative
2. For the stationary points you found, calculate the determinate D.
3. Now, optimize along the boundaries to find the absolute minimum
4. Now compile all the extrema found within the region and along each boundary, evaluate the function at each potential point.

3) The function \(z=f\left( x,y \right)=4x+5y\) is the objective function for a business process, and the function is constrained by

\(x\ge 0,y\ge 0,x+y\ge 4.y\le x+5,y\le 9,x-y\le 4\)

1. List the coordinates for the corner points.
2. Find the absolute maximum and the absolute minimum.

5) The function \(z=f\left( x,y \right)=3{{\left( x-5 \right)}^{2}}+4y+1133\) is the objective function for some process. The function is constrained by \(x+y=144\)

1. Continuous partial derivatives for z?
2. Continuous partial derivatives for g?
3. Is it possible for the partial derivatives of \(g\left( x,y \right)\) to both be zero at one point?

Create the Lagrange function \(L\left( x,y,\lambda \right)=f\left( x,y \right)-\lambda \left( g\left( x,y \right)-c \right)\), and solve the corresponding system.

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