Problem 23 - Calculus Early Transcendentals

Question

Use a graph and/or level curves to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. $$\begin{array}{l}{f(x, y)=\sin x+\sin y+\sin (x+y)} \\ {0 \leqslant x \leqslant 2 \pi, 0 \leqslant y \leqslant 2 \pi}\end{array}$$

Step-by-Step Solution

Answer

Find critical points, calculate second derivatives, and use the second derivative test to classify them.

Step 1: Understand the Function

The given function is \( f(x, y) = \sin x + \sin y + \sin(x+y) \). We need to find its local maxima, minima, and saddle points within the domain \(0 \leq x \leq 2\pi\) and \(0 \leq y \leq 2\pi\).

Step 2: Graph and Level Curves

Begin by plotting the graph of the function \(f(x, y)\) and its level curves. The graph will give a visual estimate of where maxima, minima, or saddle points might occur. Peaks in the graph represent local maxima, troughs indicate minima, and points between slopes may indicate saddle points.

Step 3: Calculate First Partial Derivatives

To find critical points, compute the first partial derivatives \( f_x \) and \( f_y \). These are:\[ f_x = \cos x + \cos(x+y) \] \[ f_y = \cos y + \cos(x+y) \] Set these derivatives equal to zero to find potential critical points.

Step 4: Solve Critical Point Equations

Solve the equations \( \cos x + \cos(x+y) = 0 \) and \( \cos y + \cos(x+y) = 0 \) simultaneously. This will give critical points, which are candidates for local maxima, minima, or saddle points.

Step 5: Calculate Second Partial Derivatives

Determine the second partial derivatives \( f_{xx}, f_{yy}, \) and \( f_{xy} \) for applying the second derivative test:\[ f_{xx} = -\sin x - \sin(x+y) \]\[ f_{yy} = -\sin y - \sin(x+y) \]\[ f_{xy} = -\sin(x+y) \]

Step 6: Apply Second Derivative Test

Employ the second derivative test. Define:\( D = f_{xx} f_{yy} - (f_{xy})^2 \). Evaluate \(D\) at each critical point:- If \( D > 0 \) and \( f_{xx} < 0 \), it's a local maximum.- If \( D > 0 \) and \( f_{xx} > 0 \), it's a local minimum.- If \( D < 0 \), it's a saddle point.

Step 7: Confirm Local Extrema and Saddle Points

After evaluating \(D\) and \(f_{xx}\) at each critical point found in Step 4, list the coordinates and nature of each point - whether they are local maxima, minima, or saddle points.

Key Concepts

Local Maximum

In multivariable calculus, a local maximum occurs at a point in the domain where the function reaches a peak compared to its neighboring points. This means that, in a small surrounding area, the function does not achieve higher values. For the function \( f(x, y) = \sin x + \sin y + \sin(x+y) \), a local maximum might appear where the graph has peaks. By calculating the partial derivatives and employing the second derivative test, we determine whether a critical point is indeed a local maximum.

To find these maxima:

• Calculate the first partial derivatives and set them to zero to find critical points.
• Use the second derivative test to confirm if a critical point is a local maximum.
• If the determinant \( D = f_{xx} f_{yy} - (f_{xy})^2 \) is positive and the second partial derivative \( f_{xx} < 0 \), the point is a local maximum.

Local Minimum

Just as a local maximum is a peak, a local minimum is a trough on the graph of a function. It is a point where the value of the function is lower than any other nearby points. In the context of our problem, which involves the function \( f(x, y) = \sin x + \sin y + \sin(x+y) \), local minima are indicated on the graph where troughs appear.

To find these minima:

• Compute the first partial derivatives \( f_x \) and \( f_y \), explore points where these derivatives equal zero.
• Apply the second derivative test: if \( D > 0 \) and \( f_{xx} > 0 \), the point is a local minimum.
• Identify and confirm these locations mathematically to substantiate graphical observations.

Saddle Point

Saddle points are points on the graph where the function does not exhibit a local maximum or minimum. Instead, they have characteristics of both. This can be observed as a point where the behavior changes between increasing and decreasing, resembling a saddle's shape.

For the function \( f(x, y) = \sin x + \sin y + \sin(x+y) \), assessing saddle points requires a careful mathematical examination such as:

• Identifying critical points by setting partial derivatives to zero.
• Applying the second derivative test: if the determinant \( D < 0 \), then the critical point is a saddle point.
• These points show convergence of increasing and decreasing trends along different axes, confirming saddle-like characteristics.

Graphing Functions

Graphing functions in multivariable calculus is a valuable skill to visually interpret behavior, such as locating local maxima, minima, and saddle points. For our function \( f(x, y) = \sin x + \sin y + \sin(x+y) \), graphing is the initial step to estimate these points intuitively.

Some tips for effective graphing include:

• Using color-coded level curves to differentiate regions with different values.
• Examining the graph for peaks, troughs, and transitional points, which likely correspond to extrema and saddle points.
• Cross-referencing these visual estimates with mathematical calculations to verify important points on the function.

Partial Derivatives

Partial derivatives are pivotal in multivariable calculus for understanding a function's behavior with respect to each variable individually. For a function like \( f(x, y) = \sin x + \sin y + \sin(x+y) \), partial derivatives help locate critical points where the function might have extrema or saddle points.

Here is how partial derivatives contribute to analyzing our function:

• Calculate \( f_x \) and \( f_y \), the partial derivatives with respect to \( x \) and \( y \).
• These derivatives afford insight into the function's slope and direction changes.
• Setting the partial derivatives to zero finds critical points, necessitating further tests to qualify them as maxima, minima, or saddle points.