Solve Highest/Lowest Points on Curve of Intersection with Lagrange Multipliers

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SUMMARY

The discussion focuses on using Lagrange multipliers to determine the highest and lowest points on the curve of intersection between an elliptic paraboloid defined by the equation z = x^2 + 2*x + 4*y^2 and a right circular cylinder described by x^2 + y^2 = 1. The critical points identified are (2/3, -√5/3) and (-√2/6, 1). Participants clarified the correct formulation of the elliptic paraboloid, confirming it as z = x^2 + 2*x + 4*y^2.

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  • Understanding of Lagrange multipliers
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  • Knowledge of right circular cylinders
  • Basic calculus concepts, particularly optimization techniques
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  • Explore the geometric interpretation of critical points on surfaces
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Students and professionals in mathematics, particularly those studying multivariable calculus, optimization techniques, and geometric analysis of surfaces.

Cherizzle
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Hi There I would like help on a question about Lagrange multipliers.

Question: Consider the intersection of two surfaces: an elliptic paraboloid z=x^2 + 2*x + 4*y^2 and a right circular cylinder x^2 + y^2 = 1. Use Lagrange multipliers to find the highest and lowest points on the curve of the intersection.

What I have so far:
I managed to find my critical points using lagrange multipliers. But now I don't know how to describe whether my points are at maximum or minimum...
The points I found were: (2/3, -\sqrt{5}/3) and (-\sqrt{2}/6, 1)
 
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Hi Cherizzle. Are you sure that's supposed to be z=x^2 + 2*x + 4*y^2 and not z=x^2 + 2*x*y + 4*y^2
 

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