The paraboloid z = x² + y² intersects the plane x + y + z = 1 in the shape of an ellipse.The objective of this problem is to find the point on the ellipse closest and furtherest away from the origin.To this end, we establish the following constrained optimization problem:• New objective: g(x, y)• New constraint:max./min. f(x, y, z) = √√x² + y² + z²z = x² + y²x+y+z=1To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint intothe objective function and square the result, then substitute the first constraint into the second constraint.=s.t.We have now constructed an equivalent optimization problem.

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The paraboloid z = x² + y² intersects the plane x + y + z = 1 in the shape of an ellipse. The objective of this problem is to find the point on the ellipse closest and furtherest away from the origin. To this end, we establish the following constrained optimization problem: max./min. f(x, y, z) = √√√x² + y² + z² z = x² + y² x+y+z=1 • New objective: g(x, y) • New constraint: s.t. To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint into the objective function and square the result, then substitute the first constraint into the second constraint.