Problem 55 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the second partial derivatives \(f_{x x^{\prime}} f_{x y^{\prime}} f_{y y^{\prime}}\) and \(f_{y x}\) at the point. $$ f(x, y)=x^{4}-3 x^{2} y^{2}+y^{2} $$

Step-by-Step Solution

Answer

The second partial derivatives of the given function are:\(f_{xx} = 12x^{2} - 6y^{2}\), \(f_{xy} = -12 xy\), \(f_{yx} = -12 xy\), \(f_{yy} = - 6x^{2} + 2\).

Step 1: Calculate the first derivative

First, calculate the first derivative with respect to x and y:\n \(f_x = 4x^{3}-6 xy^{2}\) and \(f_y = -6x^{2}y + 2y\).

Step 2: Calculate the second derivatives

Now, proceed to calculate the second derivatives based on the first derivatives:\n \(f_{xx} = \frac{∂f_x}{∂x} = 12x^{2} - 6y^{2}\), \(f_{xy} = \frac{∂f_x}{∂y} = -12 xy\), \(f_{yx} = \frac{∂f_y}{∂x} = -12 xy\) and \(f_{yy} = \frac{∂f_y}{∂y} = - 6x^{2} + 2\).

Step 3: Checking if the function is continuous

These second order partial derivatives exist for all points since the initial function f(x,y) is polynomial. Therefore it's everywhere continuous and differentiable.

Key Concepts

Polynomial Functions

Polynomial functions are a type of mathematical expression that consists of variables raised to positive integer powers. These functions can have multiple terms, such as constants, coefficients, and variables. In our exercise, we have the polynomial function:\[f(x, y) = x^4 - 3x^2y^2 + y^2\]This function involves several polynomial terms:

• A quartic term: \(x^4\)
• A mixed term: \(-3x^2y^2\)
• A quadratic term: \(y^2\)

Polynomial functions are everywhere continuous and differentiable. This means we can calculate their derivatives at any point on the function's domain. Knowing how to work with polynomial functions is key in solving more complex calculus problems.

Calculus Problems

Calculus problems can cover a variety of topics, including limits, derivatives, integrals, and differential equations. In this exercise, we focus on finding second partial derivatives, a critical element in multivariable calculus.When solving calculus problems, it's important to follow these general steps:

• Understand the function involved. Here, we have a polynomial function of two variables, \(x\) and \(y\).
• Start by finding the first partial derivatives. These give us the rate of change of the function with respect to one variable.
• Proceed to the second partial derivatives to examine more intricate behavior, including concavity, and potential critical points.

Calculating derivatives can help understand how a function behaves and reacts to changes in the input variables.

Partial Differentiation

Partial differentiation is a technique used in calculus to find the derivative of a multivariable function with respect to one variable while keeping the others constant. This is particularly useful when working with functions like \(f(x, y) = x^4 - 3x^2y^2 + y^2\), where we need to differentiate with respect to both \(x\) and \(y\).To perform partial differentiation:

• Compute the first partial derivatives: \(f_x\) and \(f_y\). These describe the function's rate of change with respect to \(x\) and \(y\), respectively.
• From the first derivatives, calculate the second partial derivatives: \(f_{xx}\), \(f_{xy}\), \(f_{yx}\), and \(f_{yy}\). These give insight into the function's curvature.

In our solution, we found:\[ f_{xx} = 12x^2 - 6y^2 \quad \text{and} \quad f_{yy} = -6x^2 + 2\]These second derivatives help determine how the slope of the tangent plane changes, shedding light on the function's deeper structure. Partial differentiation is an essential skill in understanding and analyzing complex systems in multivariable calculus.