Maximum and Minimum Values of hx y sin x sin y sin(xy)

In this article, we will explore the maximum and minimum values of the function hx y sin x sin y sin(xy). We will analyze this function using partial derivatives, critical points, and boundary conditions, and then create visual representations to confirm our findings.

1. Finding the Critical Points

First, we will compute the partial derivatives of the function with respect to x and y, and then set them to zero to find the critical points.

Partial Derivative with Respect to x:

(h_x cos x cos(xy))

Partial Derivative with Respect to y:

(h_y cos y cos(xy))

2. Setting the Partial Derivatives to Zero

Setting (h_x 0) and (h_y 0), we obtain:

[cos x cos(xy) 0] [cos y cos(xy) 0]

From these equations, we can express cos(xy) in terms of cos x and cos y:

(cos(xy) -cos x) (cos(xy) -cos y)

Equating these expressions, we get:

(-cos x -cos y Rightarrow cos x cos y)

3. Solving for x and y

The solutions to cos x cos y can be written as:

x y 2kpi x -y 2kpi for some integer k

4. Evaluating the Function at Critical Points

To evaluate the function, consider the case where x y:

h(x, x) sin x sin x sin 2x 2sin x sin 2x Using the double angle identity sin 2x 2 sin x cos x:

h(x, x) 2sin x (2sin x cos x) 2sin x (1 - cos^2 x)

5. Analyzing the Boundaries

The maximum and minimum values of sin x and sin y from -1 to 1 are:

hleft(frac{pi}{2}, frac{pi}{2}right) 1 1 0 2 hleft(frac{3pi}{2}, frac{3pi}{2}right) -1 -1 0 -2

Therefore, the maximum value is 2 and the minimum value is -2.

6. Visualizing the Function with Graphics

Making use of software tools like Mathematica 12.0, we can visualize the function to confirm our findings. Here is the surface plot:

" alt"Surface Plot of h(x, y)">

From the surface plot, it is evident that the maxima and minima are smaller than -3 and 3. To quantify the peaks and valleys, we use contour plots centered on the origin of the xy-plane.

Here are the contour levels chosen for the plot:

(-2.575, -2.5, -0.25, 0.25, 2.5, 2.575)

The blue color represents negative values, and the tan color represents positive values.

7. Finding the Exact Peaks and Valleys

Using a digital peak finding routine, we find the maximum and minimum values:

(text{Min} -2.59808 text{ at } (x, y) (-1.0472, -1.0472)) (text{Max} 2.59808 text{ at } (x, y) (1.0472, -1.0472))

Both the positive peaks and the negative valleys repeat with a periodicity of 2pi.

Conclusion

The function hx y sin x sin y sin(xy) has a maximum value of about 2.59808 and a minimum value of about -2.59808. These values repeat every 2pi due to the periodic nature of the sine function.