Answer the Question $\mathrm { f } ( \mathrm { x } , \mathrm { y } )$

question 291

Multiple Choice

Answer the question.
-Describe the results of applying the method of Lagrange multipliers to a function f(x, y) if the points (x, y) are constrained to follow a curve g(x, y) = c that is everywhere perpendicular to the level curves of f. Assume that
Both f(x, y) and g(x, y) satisfy all the requirements and conditions for the method to be applicable.

• A) Generally, local extrema of f(x, y) occur at points on the curve g(x, y) = c where the curve becomes tangent to a level curve of f(x, y). Since the curve defined by g(x, y) = c is everywhere perpendicular to the level
• Curves of f(x, y) for this particular case, it is never tangent to a level curve, and there are no local extrema
• Along g(x,y) = c. The method of Lagrange multipliers will fail to find any local extrema since there are
• None.
• B) The results cannot be generally predicted. Specific expressions for \(\mathrm { f } ( \mathrm { x } , \mathrm { y } )\) and \(g ( \mathrm { x } , \mathrm { y } ) = \mathrm { c }\) are required.
• C) Since \(\bar { V }\) is everywhere parallel to \(\nabla g\), there will be a single local minimum and a single local maximum along \(g ( x , y ) = c\). Applying the method of Lagrange multipliers should identify the locations of these two local extrema.
• D) Since \(\nabla\) is everywhere parallel to \(\nabla g\), every point on \(g ( x , y ) = c\) is a local extremum. Applying the method of Lagrange multipliers should yield the equation \(g ( x , y ) = c\).

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