Identify any extrema of the function by recognizing its...

Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema. Use a computer algebra system to graph the function and label any extrema. $f(x, y)=x^{2}+y^{2}+2 x-6 y+6$

Key Concepts

Completing the Square

This technique involves rewriting a quadratic function as the sum or difference of a perfect square and a constant. By grouping terms and adding and subtracting appropriate constants, the function can be transformed into a form that clearly shows its vertex or center. This transformation simplifies the identification of extrema (minimum or maximum values) as it directly reveals the point at which the quadratic achieves its extreme value.

Partial Derivatives and Critical Points

Partial derivatives measure the rate of change of a multivariable function with respect to each variable independently. By setting the partial derivatives equal to zero, one obtains critical points of the function, which are candidates for local extrema or saddle points. This step is crucial in the analysis of functions of several variables when determining where the function achieves its peak or its lowest value.

Second Derivative Test (Hessian Matrix)

Once critical points are found, the second derivative test using the Hessian matrix can be applied to classify the nature of each critical point. The Hessian is a square matrix composed of the second partial derivatives. By evaluating the determinants or eigenvalues of this matrix, one can decide if a given critical point corresponds to a local minimum, local maximum, or a saddle point.

Graphical Verification

Graphing the function using a computer algebra system allows for visual confirmation of the analytical results. By plotting the surface or contour lines, one can see the position of extrema and verify that the identified critical points align with the behavior of the function. This step is valuable for intuition and for catching any miscalculations in the analytical process.

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