91Ó°ÊÓ

Optimization is a fundamental concept in mathematics where we aim to find the best solution, often the maximum or minimum, within a given set of constraints. In this exercise, we are looking for the point on the plane defined by the equation \(2x+3y+z-11=0\) that minimizes the function \(4x^2+y^2+z^2\). This means finding a set of values for \(x, y,\) and \(z\) that satisfies the plane's equation while keeping the value of our function as low as possible. By achieving this, we solve for the most 'optimal' point that meets all the conditions. This technique is crucial in numerous real-world applications like engineering and economics, where optimal resource allocation is necessary.

Multivariable calculus extends single-variable calculus techniques to functions with several variables. In this context, we deal with functions of the form \(f(x, y, z)\), where changes in one variable can affect the output in conjunction with changes in others. The given function \(4x^2+y^2+z^2\) is a classic example, where each variable influences the overall value of the function. Multivariable calculus allows us to understand and manipulate these functions to explore surfaces and curves in higher dimensions, solve equations, and find optimal points as we did in this exercise. Partial derivatives play a crucial role here, as they help us determine the rate of change of the function concerning each of its variables.

Constrained minimization involves finding the minimum value of a function subject to certain constraints. In our exercise, this means minimizing \(4x^2+y^2+z^2\) while still satisfying the plane equation \(2x+3y+z-11=0\). The Lagrange multipliers method is an effective technique to handle such problems. By introducing a new variable \(\lambda\) (called the Lagrange multiplier), we create the Lagrangian function, which helps us convert the constrained problem into an unconstrained one. We then differentiate this Lagrangian with respect to all variables (including \(\lambda\)) and solve the resulting system of equations. This method provides the optimal values and ensures that our solution respects the given constraint.

Partial derivatives measure how a multivariable function changes when one of its variables is varied while the others are held constant. They are essential in optimization problems with multiple variables. For our problem, we compute the partial derivatives of the Lagrangian \(\mathbf{L}(x, y, z, \lambda) = 4x^2 + y^2 + z^2 + \lambda (2x + 3y + z - 11)\) with respect to \(x, y, z,\) and \(\lambda\). Setting these partial derivatives to zero gives us a system of equations, which we solve to find our variables' optimal values. This systematic approach enables us to dissect complex multivariable relationships and optimize functions within constraints.