Problem 13 - Calculus with Concepts in Calculus

Question

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=x^{2}-e^{y^{2}} $$

Step-by-Step Solution

Answer

The function has a saddle point at (0, 0).

Step 1: Find the first partial derivatives

To find the critical points of the function, we first need to determine the first partial derivatives with respect to both variables, \(x\) and \(y\). The partial derivative of \(f\) with respect to \(x\) is \( f_x = \frac{\partial}{\partial x} (x^2 - e^{y^2}) = 2x \). Similarly, the partial derivative with respect to \(y\) is \( f_y = \frac{\partial}{\partial y} (x^2 - e^{y^2}) = -2ye^{y^2} \).

Step 2: Set partial derivatives to zero

Critical points occur where both first partial derivatives are equal to zero. Setting \(f_x = 0\), we get \(2x = 0\) which implies \(x = 0\). Setting \(f_y = 0\), we have \(-2ye^{y^2} = 0\). Since \(e^{y^2}\) is never zero, \(y\) must be zero for the equation to hold true. Thus, solving these, we find a critical point at \((0, 0)\).

Step 3: Find second partial derivatives

To determine the nature of the critical point, we calculate the second partial derivatives. These are: \( f_{xx} = \frac{\partial}{\partial x}(2x) = 2 \), \( f_{yy} = \frac{\partial}{\partial y}(-2ye^{y^2}) = -2e^{y^2} - 4y^2e^{y^2} \), and \( f_{xy} = f_{yx} = 0 \) since the mixed partial derivative of \(x^2 - e^{y^2}\) is zero.

Step 4: Use the second derivative test

Apply the second derivative test using the Hessian determinant \(H = f_{xx}f_{yy} - (f_{xy})^2\) at the critical point \((0, 0)\). We calculate \(H = (2)(-2e^{0}) - (0)^2 = -4\). Since \(H < 0\), the critical point \((0, 0)\) is a saddle point.

Key Concepts

Partial Derivatives

Partial derivatives are a crucial tool when dealing with multivariable functions. Essentially, they help us understand how a function changes if we alter one variable, while keeping the other constant. Let's delve into the function you've come across: \( f(x, y) = x^2 - e^{y^2} \). To explore these changes, we take the partial derivative of \( f(x, y) \) with respect to \( x \), denoted as \( f_x \), and with respect to \( y \), denoted as \( f_y \).

• For \( f_x \), you differentiate \( x^2 \) to get \( 2x \), which shows how \( f \) varies with changes in \( x \).
• For \( f_y \), differentiating \( -e^{y^2} \) yields \( -2ye^{y^2} \), capturing the effect of changes in \( y \).

Finding the points where both derivatives equal zero helps identify critical points. In this case, setting \( 2x = 0 \) leads to \( x = 0 \), and \( -2ye^{y^2} = 0 \) gives \( y = 0 \), marking \( (0,0) \) as a critical point.

Second Derivative Test

The second derivative test is a method used to ascertain the nature of a critical point identified by partial derivatives. This involves calculating the second-order derivatives and using them to understand the curvature of the function around the critical point.

• Compute the second derivative with respect to \( x \), \( f_{xx} = 2 \).
• Compute the second derivative with respect to \( y \), \( f_{yy} = -2e^{y^2} - 4y^2e^{y^2} \).
• Compute the mixed second derivative, \( f_{xy} = 0 \).

These derivatives help identify whether you have a maximum, minimum, or saddle point at \( (0,0) \).

Hessian Determinant

The Hessian determinant is essential for applying the second derivative test effectively. It is formulated as \( H = f_{xx}f_{yy} - (f_{xy})^2 \). At a critical point, this determinant helps reveal the nature of the point. To check the critical point \( (0,0) \):

• Evaluate \( H = (2)(-2e^0) - (0)^2 \).
• The calculation yields \( H = -4 \).

When \( H < 0 \), the function has a saddle point. This determinant value indicates that \( (0,0) \) is indeed a saddle point.

Saddle Point Analysis

Saddle points are unique in that they are not maxima or minima, but rather points where the function curves upward in one direction and downward in another. In the context of this problem, the critical point \( (0,0) \) is a saddle point. Here's how you can analyze a saddle point:

• Assess the behavior of the second derivatives and their influence on the surface of the function.
• Since \( H < 0 \), the function behaves like a saddle at \( (0,0) \), showcasing a point of inflection.

Saddle points imply neither an optimal value nor an endpoint, but they point to interesting changes in the function's direction.