Optimizing Convex Functions with Multivariable Calculus: A Case Study

Convex functions play a critical role in various fields of mathematics and its applications. One such case is the given surface defined by the equation z x^2 - 4xy 6y^2 - 4y 4. This article delves into the process of finding the minimum value of this function using multivariable calculus techniques.

Understanding the Surface: Convexity and Minimum Value

The surface described by the equation z x^2 - 4xy 6y^2 - 4y 4 is a convex elliptic paraboloid, meaning it is bowl-shaped and has a single global minimum. This minimum is precisely where the vertex of the paraboloid is located. Convexity ensures that any local minimum is also a global minimum, making the process straightforward.

Finding the Minimum Value Via Partial Derivatives

To find the minimum value of the function, we first need to find the critical points. This involves setting the partial derivatives equal to zero:

The Partial Derivatives

The partial derivatives of the function are:

partial_x z 2x - 4y 0

partial_y z -4x 12y - 4 0

These equations can be simplified as:

x - 2y 0

-x 3y - 1 0

By subtracting the first equation from the second, we obtain:

y 1

Substituting y 1 back into the first equation, we get:

x - 2(1) 0, leading to x 2.

Verification at the Critical Point

The value of the function at the critical point (x, y) (-2, 1) can be computed directly:

z (-2)^2 - 4(-2)(1) 6(1)^2 - 4(1) 4 4 8 6 - 4 4 18 - 4 2

This result aligns with our expectations and confirms that the minimum value of the function is 2.

Finding the Function as a Sum of Squares

To further validate the result, we can express the function as a sum of squares. By focusing on the terms involving x, we can rewrite the function as:

x^2 - 4xy 6y^2 - 4y 4 x^2 - 4xy 4y^2 2y^2 - 4y 4

This simplifies to:

(x - 2y)^2 2(y^2 - 2y 1) (x - 2y)^2 2(y - 1)^2

The minimum value of both square terms (x - 2y)^2 and 2(y - 1)^2 is 0, and this occurs when (x, y) (-2, 1). Therefore, the minimum value of the function is indeed 2.

Conclusion and Further Applications

Our analysis demonstrates that the minimum value of the function z x^2 - 4xy 6y^2 - 4y 4 is 2, occurring at the point (-2, 1). This example provides insight into the power of multivariable calculus in solving optimization problems. Understanding such techniques is crucial for a wide range of applications, including economics, engineering, and data science.

Further Reading

For more detailed applications of multivariable calculus, see the following resources:

Applications of Multivariable Calculus Multivariable Calculus in Finance Understanding Convex Functions with Multivariable Calculus

Understanding these concepts will provide a strong foundation for tackling more complex optimization problems.