Problem 15 - 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. $$ k(x, y)=e^{x} \sin y $$

Step-by-Step Solution

Answer

No critical points exist for the function.

Step 1: Find Partial Derivatives

To find the critical points of the function \( k(x, y) = e^{x} \ sin(y) \), we first need to compute the partial derivatives with respect to \( x \) and \( y \). The partial derivative with respect to \( x \) is: \[ \frac{\partial k}{\partial x} = e^{x} \ sin(y). \]The partial derivative with respect to \( y \) is: \[ \frac{\partial k}{\partial y} = e^{x} \ cos(y). \]

Step 2: Set Partial Derivatives to Zero

To find the critical points, we set the partial derivatives equal to zero:1. \( e^{x} \ sin(y) = 0 \) implies \( sin(y) = 0 \).2. \( e^{x} \ cos(y) = 0 \) implies \( cos(y) = 0 \). Since \( e^{x} \) is never zero, we focus on the trigonometric equations. The solutions to these are:- \( sin(y) = 0 \) when \( y = n\pi \) for integer \( n \).- \( cos(y) = 0 \) when \( y = \frac{\pi}{2} + m\pi \) for integer \( m \).

Step 3: Determine Intersection of Solutions

The critical points occur when both conditions are met simultaneously.However, \( sin(y) = 0 \) and \( cos(y) = 0 \) cannot be true at the same \( y \). Therefore, there are no real values of \( y \) that satisfy both conditions simultaneously.

Step 4: Analyze Critical Points

Since there are no common solutions for both partial derivatives simultaneously being zero, there are no critical points. Thus, the function \( k(x, y) = e^{x} \ sin(y) \) does not have any relative maxima, minima, or saddle points.

Key Concepts

Partial Derivatives

Partial derivatives are a crucial tool in multivariable calculus for understanding how a function changes as one of the variables changes, while the others are kept constant. For a function of two variables, say \( f(x, y) \), the partial derivative with respect to \( x \) involves differentiating \( f(x, y) \) with respect to \( x \), treating \( y \) as a constant.

• To find critical points in multivariable functions, calculate the partial derivatives for each variable.
• These derivatives provide the slope of the tangent line in the direction of each variable.

In the given function \( k(x, y) = e^{x} \, \sin(y) \), the partial derivatives are:

• \( \frac{\partial k}{\partial x} = e^{x} \sin(y) \)
• \( \frac{\partial k}{\partial y} = e^{x} \cos(y) \)

These expressions help identify how the function changes in response to small changes in \( x \) and \( y \). Setting these derivatives to zero is the key step to finding potential critical points.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in solving calculus problems involving periodic behavior. In this exercise, we need to solve equations involving \( \sin(y) \) and \( \cos(y) \) to find critical points.

• \( \sin(y) = 0 \) implies \( y = n\pi \), where \( n \) is an integer.
• \( \cos(y) = 0 \) implies \( y = \frac{\pi}{2} + m\pi \), where \( m \) is an integer.

Both conditions cannot be true simultaneously for any real \( y \). This is because the solutions for these equations do not overlap:

• when the sine function is zero, the cosine is either 1 or -1, and vice versa.
• Critical points would need a common \( y \) value satisfying both equations, which doesn't exist.

Thus, solving trigonometric equations is essential in identifying points that possibly define extrema and saddle points.

Relative Maxima and Minima

Determining relative maxima, minima, and saddle points helps understand the behavior of multivariable functions. Relative extrema refer to points where a function achieves a local maximum or minimum value.

• A **relative maximum** occurs where the function transitions from increasing to decreasing.
• A **relative minimum** occurs where the function transitions from decreasing to increasing.

Saddle points are unique as they are neither maxima nor minima but are still critical points where the derivative is zero in all directions. In the case of \( k(x, y) = e^{x} \sin(y) \):

• The analysis revealed no common \( y \) solutions, meaning no critical points exist where both partial derivatives are zero simultaneously.
• Hence, no relative maxima, minima, or saddle points are present for this function.

Understanding how critical points relate to the shape of a graph helps in grasping the full picture of a function's behavior.