Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=x^{3}+x^{2} y^{2}+y^{3}+x+y\)

Step-by-Step Solution

Answer

In summary, the second-order partial derivatives for the given function are: \(f_{xx} = 6x\), \(f_{yy} = 6y\), and the mixed second-order partial derivatives \(f_{xy} = f_{yx} = 4xy\).

Step 1: Find the first-order partial derivatives

First, we find the partial derivatives of \(f(x, y)\) with respect to both \(x\) and \(y\). For partial derivative with respect to \(x\) (\(f_x\)): \[f_x = \frac{\partial}{\partial x}(x^3 + x^2y^2 + y^3 + x + y) = 3x^2 + 2xy^2 + 1\] For partial derivative with respect to \(y\) (\(f_y\)): \[f_y = \frac{\partial}{\partial y}(x^3 + x^2y^2 + y^3 + x + y) = 2x^2y + 3y^2 + 1\]

Step 2: Find the second-order partial derivatives

Now, we find the second-order partial derivatives \(f_{xx}\), \(f_{yy}\), \(f_{xy}\), and \(f_{yx}\) by taking the partial derivatives of \(f_x\) and \(f_y\) with respect to \(x\) and \(y\). For second-order partial derivative with respect to \(x\) (\(f_{xx}\)): \[f_{xx} = \frac{\partial^2}{\partial x^2}(f) = \frac{\partial}{\partial x}(3x^2 + 2xy^2 + 1) = 6x\] For second-order partial derivative with respect to \(y\) (\(f_{yy}\)): \[f_{yy} = \frac{\partial^2}{\partial y^2}(f) = \frac{\partial}{\partial y}(2x^2y + 3y^2 + 1) = 6y\] For second-order mixed partial derivative (\(f_{xy}\)): \[f_{xy} = \frac{\partial^2}{\partial x \partial y}(f) = \frac{\partial}{\partial x}(2x^2y + 3y^2 + 1) = 4xy\] For second-order mixed partial derivative (\(f_{yx}\)): \[f_{yx} = \frac{\partial^2}{\partial y \partial x}(f) = \frac{\partial}{\partial y}(3x^2 + 2xy^2 + 1) = 4xy\]

Step 3: Show that the mixed partial derivatives are equal

Finally, we compare the mixed partial derivatives \(f_{xy}\) and \(f_{yx}\) to show that they are equal: \[f_{xy} = 4xy\] \[f_{yx} = 4xy\] Since \(f_{xy}\) and \(f_{yx}\) are equal, we have proven that the mixed partial derivatives of the given function are equal.

Key Concepts

Partial Derivatives

Partial derivatives are an essential concept in calculus, especially when working with functions of several variables. For a function of two variables, say \(f(x, y)\), partial derivatives help us understand how the function changes as each variable changes while keeping the other constant.

To find a partial derivative concerning \(x\), denoted as \(f_x\), we treat \(y\) as a constant and differentiate \(f\) with respect to \(x\). Conversely, to find the partial derivative with respect to \(y\), denoted as \(f_y\), we treat \(x\) as a constant and differentiate \(f\) with respect to \(y\). These derivatives help gauge the rate of change of the function in each direction. Here's a simple view on how to calculate them:

• When finding \(f_x\), ignore any terms without \(x\) as they become zero, and follow your usual differentiation rules for the remaining terms.
• When finding \(f_y\), ignore any terms without \(y\) and differentiate with respect to \(y\).

Practically, these calculations help in various fields, including physics and engineering, to analyze systems that depend on multiple inputs.

Mixed Partial Derivatives

Mixed partial derivatives take the concept of partial derivatives a step further by exploring how a function changes when both variables change simultaneously. For our function \(f(x, y)\), we are interested in derivatives like \(f_{xy}\) and \(f_{yx}\).

To compute the mixed partial derivative \(f_{xy}\), you first find \(f_x\), the partial derivative with respect to \(x\), and then differentiate \(f_x\) with respect to \(y\). Similarly, for \(f_{yx}\), you first find \(f_y\) and then differentiate this expression with respect to \(x\). This process gives you two second-order mixed derivatives:

• \(f_{xy} = \frac{\partial^2 f}{\partial y \partial x}\)
• \(f_{yx} = \frac{\partial^2 f}{\partial x \partial y}\)

A crucial property here is that under certain conditions, the mixed partial derivatives are equal, that is, \(f_{xy} = f_{yx}\). This equality, known as the symmetry of second derivatives, holds under the assumption that the mixed derivatives are continuous, which is often the case for well-behaved functions.

Mathematical Proof

Proving that mixed partial derivatives are equal is a key exercise in mastering calculus. The proof typically involves showing that \(f_{xy}\) equals \(f_{yx}\), as indicated by the original exercise.

The process involves:

• Calculating \(f_{xy}\) by differentiating \(f_x\) with respect to \(y\).
• Calculating \(f_{yx}\) by differentiating \(f_y\) with respect to \(x\).

If both calculations result in the same expression, we conclude that \(f_{xy} = f_{yx}\). This proof uses the concept of Clairaut's Theorem, which states that if the second mixed partial derivatives are continuous in a neighborhood of a point, then they are equal at that point.

Checking this property becomes essential in understanding the behavior of functions of multiple variables and is widely applicable in optimization problems and modeling real-world phenomena, where variables interact in complex ways.