Problem 2 - Calculus for Biology and Medicine

Question

The functions are defined for all \((x, y) \in \boldsymbol{R}^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). \(f(x, y)=-2 x^{2}-y^{2}+3 y\)

Step-by-Step Solution

Answer

The function has a local maximum at \((0, \frac{3}{2})\).

Step 1: Determine Critical Points

To find candidates for local extrema, we first find the partial derivatives of \(f(x,y)\): \(f_{x}(x,y) = \frac{\partial f}{\partial x} = -4x\) and \(f_{y}(x,y) = \frac{\partial f}{\partial y} = -2y + 3\). Set these derivatives equal to zero to find critical points:1. \(-4x = 0 \) implies \( x = 0 \).2. \(-2y + 3 = 0 \) implies \( y = \frac{3}{2} \).Thus, the only critical point is \((0, \frac{3}{2})\).

Step 2: Calculate the Hessian Matrix

The Hessian matrix \(H(f)\) is defined as:\[H(f) = \begin{bmatrix}\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \\end{bmatrix}\]Calculate the second partial derivatives:- \(\frac{\partial^2 f}{\partial x^2} = -4\)- \(\frac{\partial^2 f}{\partial y^2} = -2\)- \(\frac{\partial^2 f}{\partial x \partial y} = 0\)- \(\frac{\partial^2 f}{\partial y \partial x} = 0\)The Hessian is:\[H(f) = \begin{bmatrix}-4 & 0 \0 & -2 \\end{bmatrix}\]This matrix will be used to determine the nature of the critical point.

Step 3: Apply Second Derivative Test Using Hessian Matrix

The determinant of the Hessian matrix is calculated as:\[\text{det}(H(f)) = (-4)(-2) - (0)(0) = 8\]Since \(\text{det}(H(f)) > 0\) and \(\frac{\partial^2 f}{\partial x^2} = -4 < 0\), according to the second derivative test, the point \((0, \frac{3}{2})\) is a local maximum.

Key Concepts

Hessian Matrix

The Hessian matrix is a crucial tool in multivariable calculus. It helps in analyzing the behavior of functions of several variables to determine extrema, like maxima and minima. In simpler terms, if you have a surface defined by a function, the Hessian matrix helps us understand how the surface curves at different points.

The Hessian matrix is composed of second order partial derivatives of a function. For a function \( f(x, y) \), the Hessian matrix \( H(f) \) is defined as:

• The top left element \( \frac{\partial^2 f}{\partial x^2} \), is the second partial derivative with respect to \( x \).
• The top right and bottom left elements, \( \frac{\partial^2 f}{\partial x \partial y} \) and \( \frac{\partial^2 f}{\partial y \partial x} \), reflect the cross derivative.
• The bottom right element \( \frac{\partial^2 f}{\partial y^2} \), is the second partial derivative with respect to \( y \).

By calculating these, you form the Hessian matrix, which can then be used to assess the nature of critical points found in the function.

Critical Points

To understand the behavior of a function, locating its critical points is essential. Critical points occur where the first partial derivatives of the function are zero or undefined. By finding where these derivatives equate to zero, we identify possible points where the function may have a local maximum, minimum, or saddle point.

For instance, take the function \( f(x, y) \). We find its partial derivatives, \( f_x \) and \( f_y \), and set them equal to zero:

• \( f_x = 0 \): Solutions to this equation provide potential \( x \) values for critical points.
• \( f_y = 0 \): Solutions to this equation give potential \( y \) values.

By solving these, you obtain the coordinates of the critical points. These points become candidates for further testing using the Hessian matrix to determine their nature.
Critical points can be thought of as important landmarks on the graph, indicating where significant changes in the function's behavior might happen.

Second Derivative Test

Once you have the Hessian matrix and critical points, the Second Derivative Test is your next step. This test helps decide whether each critical point is a local maximum, local minimum, or a saddle point.

The procedure involves:

• Calculating the determinant of the Hessian matrix, denoted \( \text{det}(H) \).
• If \( \text{det}(H) > 0 \):
  • and \( \frac{\partial^2 f}{\partial x^2} > 0 \), the point is a local minimum.
  • and \( \frac{\partial^2 f}{\partial x^2} < 0 \), the point is a local maximum.
• If \( \text{det}(H) < 0 \), the critical point is a saddle point.
• If \( \text{det}(H) = 0 \), the test is inconclusive.

In essence, the Second Derivative Test determines how the surface around a critical point curvatures, thus informing us about the local properties of the function at these points.