Problem 10 - Calculus for Biology and Medicine

Question

The functions are defined for all \((x, y) \in R^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). $$ f(x, y)=y \sin x $$

Step-by-Step Solution

Answer

The function has saddle points at \((x, y) = (n\pi, 0)\) where \( n \) is any integer.

Step 1: Find Partial Derivatives

First, compute partial derivatives of the function with respect to both variables. The partial derivative of the function with respect to \( x \) is \( f_x = y \, \cos x \) and with respect to \( y \) is \( f_y = \sin x \).

Step 2: Set Partial Derivatives to Zero

To find critical points, set the partial derivatives equal to zero: \( y \, \cos x = 0 \) and \( \sin x = 0 \). This gives two conditions: \( y = 0 \) or \( \cos x = 0 \), and \( \sin x = 0 \).

Step 3: Solve the System of Equations

Start with \( \sin x = 0 \), which implies \( x = n\pi \) where \( n \) is any integer. For \( \cos x = 0 \), \( x = (2m+1)\frac{\pi}{2} \) where \( m \) is any integer. Therefore, critical points are \((x, y) = (n\pi, 0)\) or \((x, y) = ((2m+1)\frac{\pi}{2}, y)\).

Step 4: Construct Hessian Matrix

The Hessian matrix \( H \) is given by: \[ H = \begin{bmatrix} f_{xx} & f_{xy} \ f_{yx} & f_{yy} \end{bmatrix} \] where \( f_{xx} = -y \sin x \), \( f_{xy} = \cos x \), \( f_{yx} = \cos x \), and \( f_{yy} = 0 \).

Step 5: Evaluate Hessian at Critical Points

For critical points \((n\pi, 0)\), the Hessian simplifies to \( \begin{bmatrix} 0 & (-1)^n \ (-1)^n & 0 \end{bmatrix} \). The determinant is \( (-1)^{2n} - 0 \) which equals 0, indicating a saddle point. Since only zeros are in \( f_y \) derivatives, attempt for other points will also lead to saddle points.

Step 6: Conclusion on Type of Critical Points

The determinant of the Hessian is zero at all critical points, indicating they all behave like saddle points, as there is no definite positive or negative definiteness to classify them as maxima or minima.

Key Concepts

Local Extrema

Local extrema refer to the points within a region on a graph where a function reaches either a minimum or a maximum value. Think of it like finding the hills or valleys on a curve. In mathematical terms, a local minimum is where the function value is lower than its surroundings, while a local maximum is higher than its neighboring points. For functions of two variables, like in our exercise, these are points \(x, y\) where the slope of the tangent plane is zero.

To determine the nature of these critical points, we often use calculus tools like the Hessian matrix. It helps classify the local extrema not just as maximum or minimum, but also indicates if the critical point could be a saddle point where the function changes direction. To better understand this, imagine standing on a saddle of a horse; it's neither the highest nor the lowest point, but you can move up or down depending on the direction.

Partial Derivatives

Partial derivatives are a key concept when analyzing functions with more than one variable. They represent the rate at which the function changes as one of the variables varies, while all other variables are kept constant. Think of it as a derivative that focuses on only one direction at a time.

In the context of finding critical points, partial derivatives are crucial because they lead us to conditions that identify where slopes vanish. Partial derivatives with respect to \(x\) and \(y\) are used to form what are called the first derivative tests. By setting these derivatives equal to zero, as shown in the solution, we can locate potential points for local extrema or saddle points.

• Partial derivative with respect to \(x\) is expressed as \(f_x = y \cos x\), describing how the function changes as \(x\) changes.
• Partial derivative with respect to \(y\) is \(f_y = \sin x\), describing changes as \(y\) changes.

When both these derivatives equal zero, we're at a critical point, requiring further tools like the Hessian for classification.

Critical Points

Critical points are locations on the surface of a multivariable function where the tangent plane is flat. This means that the function's slope is zero in all directions at that point. To find critical points, we set the partial derivatives equal to zero and solve the resulting equations.

In our exercise, solving \(y \cos x = 0\) and \(\sin x = 0\) delivers critical points like \( (n\pi, 0) \) and \( ((2m+1)\frac{\pi}{2}, y) \). Each critical point needs evaluation to know its nature: Is it a point of maximum, minimum, or neither (a saddle point)? Here, we rely on the Hessian matrix.

The Hessian matrix helps analyze these points based on its determinant. If the determinant is zero, like in our exercise, it indicates a saddle point. Saddle points are neither solely upward peaks nor downward valleys but represent a change in curvature across the point. Thus, when locating critical points, we always need to complete the story with further investigation using tools like the Hessian matrix.