Problem 7 - Applied Calculus

Question

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{2}+x y+3 y$$

Step-by-Step Solution

Answer

The critical point is (-3, 6), which is a saddle point.

Step 1: Find the First Partial Derivatives

To find the critical points, we first need the partial derivatives of the function with respect to both variables. For the function \( f(x, y) = x^2 + xy + 3y \), the first partial derivatives are:\[ f_x = \frac{\partial}{\partial x}(x^2 + xy + 3y) = 2x + y \]\[ f_y = \frac{\partial}{\partial y}(x^2 + xy + 3y) = x + 3 \]

Step 2: Set Partial Derivatives to Zero

The critical points occur where the first partial derivatives are zero. Solve the equations:1. \( 2x + y = 0 \)2. \( x + 3 = 0 \)

Step 3: Solve the System of Equations

From the second equation, solve for \(x\):\[ x + 3 = 0 \Rightarrow x = -3 \]Substitute \( x = -3 \) into the first equation:\[ 2(-3) + y = 0 \Rightarrow y = 6 \]The critical point is \((-3, 6)\).

Step 4: Find the Second Partial Derivatives and Hessian

To determine the nature of the critical point, calculate the second partial derivatives:\[ f_{xx} = 2, \quad f_{yy} = 0, \quad f_{xy} = 1 \]Calculate the Hessian determinant:\[ H = f_{xx}f_{yy} - (f_{xy})^2 = (2)(0) - (1)^2 = -1 \]

Step 5: Determine the Nature of the Critical Point

The Hessian determinant \( H = -1 \) is less than zero. When the Hessian is negative, the critical point is a saddle point.

Key Concepts

Partial Derivatives

Partial derivatives are a foundational tool in multivariable calculus, allowing us to understand how a function changes as one of the variables changes while keeping the others constant. It's like peeking into just one direction at a time in a multidimensional landscape.
To find the critical points of a function like \( f(x, y) = x^2 + xy + 3y \), we start by calculating the partial derivatives with respect to \( x \) and \( y \). These derivatives tell us how the function \( f \) changes in response to slight variations in these variables. In essence, we are measuring the slope in the \( x \) and \( y \) directions.

• For \( x \), the partial derivative \( f_x \) indicates how \( f \) changes per unit change in \( x \) alone.
• Similarly, \( f_y \) signals the rate of change of \( f \) per unit change in \( y \).

By setting these partial derivatives to zero, we find the critical points where the surface is flat in both directions—potential peaks, valleys, or saddle points.

Hessian Determinant

The Hessian determinant is like a compass that helps us assess the type of critical point we have found by examining the curvature of the function's graph around that point. It involves the second partial derivatives, giving us deeper insights into how the function behaves.
In our example, the Hessian determinant is calculated as \( H = f_{xx}f_{yy} - (f_{xy})^2 \). Here is a breakdown:

• \( f_{xx} \): the second partial derivative with respect to \( x \), showing how the slope in the \( x \) direction changes as \( x \) changes.
• \( f_{yy} \): the second partial derivative with respect to \( y \), indicating changes in the slope in the \( y \) direction.
• \( f_{xy} \): the mixed partial derivative, revealing how changes in \( x \) affect the slope in the \( y \) direction and vice versa.

The value of the Hessian tells us if the critical point is a local extremum or a saddle point. A negative Hessian, as in this exercise, indicates a saddle point.

Saddle Point

A saddle point is a fascinating concept in calculus. It might seem strange because it's neither a peak nor a valley, but rather a point where the surface changes direction.
Unlike a hill or a pit, where all points around are either higher or lower, a saddle point has a mix: some directions look like a peak, others like a valley. This happens when the Hessian determinant is negative.
In our example, the critical point \((-3, 6)\) was determined to be a saddle point because the Hessian was \(-1\). So, the graph of \( f(x, y) \) at \((-3, 6)\) curves upward in one direction and downward in another. It's like the saddle on a horse, which has an upward curve from front to back and a downward curve from side to side.

Second Partial Derivatives

Second partial derivatives provide information about the curvature of the function's graph. They tell us about the rate of change of the slopes found using first partial derivatives.
Let's consider them for \( f(x, y) = x^2 + xy + 3y \):

• \( f_{xx} = 2 \): shows how the slope changes in the \( x \) direction as \( x \) varies.
• \( f_{yy} = 0 \): indicates there's no change in the slope in the \( y \) direction as \( y \) changes; the surface is flat in this direction.
• \( f_{xy} = 1 \): describes the interaction of changes between \( x \) and \( y \).

These second partial derivatives help build the Hessian matrix, which is crucial for deciding the nature of the critical points. In practice, they provide a clearer picture of how the function behaves around these important points.