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.

Question

A. The results cannot be generally predicted. Specific expressions for f(x, y) and g(x, y) = c are required.
B. 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.
C. Since ?f is everywhere parallel to ?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 ?f is everywhere parallel to ?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.

Answer 1

Answer: B