Problem 12 - Calculus for Biology and Medicine

Question

Consider the function $$ f(x, y)=a x^{2}+b y^{2} $$ (a) Show that $$ \nabla f(0,0)=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] $$ (b) Find values for \(a\) and \(b\) such that (i) \((0,0)\) is a local minimum, (ii) \((0,0)\) is a local maximum, and (iii) \((0,0)\) is a saddle point.

Step-by-Step Solution

Answer

(a) The gradient at (0,0) is [0,0]. (b) Local min: a>0, b>0; Local max: a<0, b<0; Saddle point: a>0, b<0 or a<0, b>0.

Step 1: Calculate the Gradient of f(x,y)

The gradient of a function \( f(x, y) \) is given by \( abla f(x, y) = \left[ \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right] \). First, compute the partial derivatives:- \( \frac{\partial f}{\partial x} = 2ax \)- \( \frac{\partial f}{\partial y} = 2by \).Thus, the gradient is \( abla f(x, y) = \left[ 2ax, 2by \right] \).

Step 2: Evaluate the Gradient at (0,0)

Substitute \( x = 0 \) and \( y = 0 \) into the gradient found in Step 1:- \( 2a \cdot 0 = 0 \).- \( 2b \cdot 0 = 0 \).Therefore, \( abla f(0,0) = \left[ 0, 0 \right] \). This confirms part (a) of the problem.

Step 3: Identify Conditions for Local Extrema or Saddle Point

The second derivative test for functions of two variables involves the Hessian matrix, \( H \), given by:\[H = \begin{bmatrix}\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2}\end{bmatrix}\]Compute the second derivatives:- \( \frac{\partial^2 f}{\partial x^2} = 2a \).- \( \frac{\partial^2 f}{\partial y^2} = 2b \).- \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} = 0 \).Thus, the Hessian becomes:\[H = \begin{bmatrix}2a & 0 \0 & 2b\end{bmatrix}\]

Step 4: Determine Values for a and b

To classify the critical point \((0,0)\):- **Local Minimum**: \( (0,0) \) is a local minimum if \( 2a > 0 \) and \( 2b > 0 \). For simplicity, set \( a > 0 \) and \( b > 0 \).- **Local Maximum**: \( (0,0) \) is a local maximum if \( 2a < 0 \) and \( 2b < 0 \). Set \( a < 0 \) and \( b < 0 \).- **Saddle Point**: \((0,0)\) is a saddle point if the determinant of \( H \), \( (2a)(2b) - 0^2 = 4ab \), is negative. Choose \( a > 0 \) and \( b < 0 \), or vice versa.

Key Concepts

Partial Derivatives

In calculus, a partial derivative of a function with several variables is its derivative with respect to one of those variables while keeping the others constant. This concept is crucial in multivariable calculus. For the function given, \( f(x, y)=a x^{2}+b y^{2} \), we use partial derivatives to find how the function changes as \( x \) or \( y \) changes, one at a time without altering the other.

• To find the partial derivative with respect to \( x \), denoted as \( \frac{\partial f}{\partial x} \), consider \( y \) as a constant. This gives us \( 2ax \).
• Similarly, for the partial derivative with respect to \( y \), \( \frac{\partial f}{\partial y} \), treat \( x \) as constant, resulting in \( 2by \).

These partial derivatives show how \( f \) increases or decreases as we change \( x \) and \( y \) individually. Knowing partial derivatives is essential for further concepts like gradients and Hessian matrices.

Gradient

The gradient is a vector containing the first partial derivatives of a function with multiple variables. It points in the direction of the steepest ascent of the function. For a function \( f(x, y) \), the gradient is defined as \( abla f(x, y) = \left[ \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right] \).In our case, both components of the gradient, \( \frac{\partial f}{\partial x} = 2ax \) and \( \frac{\partial f}{\partial y} = 2by \), indicate how the function changes in the respective variable direction. At the point \((0,0)\), substituting x and y into the gradient yields \( abla f(0,0) = [0,0] \), implying no change in any direction, which is important in understanding extrema.The gradient's role is pivotal in optimization as it helps identify critical points, like maxima, minima, or saddle points. Its magnitude gives the steepness of the slope, while its direction indicates the path of greatest increase.

Hessian Matrix

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function, crucial for determining the nature of critical points in multivariable functions. For a function \( f(x, y) \), the Hessian matrix \( H \) is constructed as:\[H = \begin{bmatrix}\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2}\end{bmatrix}\]For our function, these second derivatives simplify to \( H = \begin{bmatrix}2a & 0 \ 0 & 2b\end{bmatrix} \). The diagonal form and zeros imply no mixed partial derivative influence.To decide if \((0,0)\) is a local minimum, maximum, or saddle point, we use the determinant of the Hessian, \( 4ab \):

• A positive \( 4ab \) with both \( 2a > 0 \) and \( 2b > 0 \) indicate a local minimum.
• Negative \( 4ab \) means a saddle point.
• If \( 2a < 0 \) and \( 2b < 0 \), it signifies a local maximum.

This analysis is critical in applications such as optimization problems and economic models where determining the nature of critical points is vital.