Determine all the absolute extreme values for the function f(x, y) = sin x sin y sin(x + y) on the square
0 <x <PI ,0 < y < PI.
To find the absolute extreme values of the function f(x, y) = sin(x) sin(y) sin(x + y) on the given square (0 < x < PI, 0 < y < PI), we need to examine the critical points and the boundary points of the region.
1) Critical Points:
To find the critical points, we need to take the partial derivatives of f(x, y) with respect to x and y, and then set them equal to zero.
∂f/∂x = cos(x) sin(y) sin(x + y) + sin(x) sin(y) cos(x + y) = 0 (Equation 1)
∂f/∂y = sin(x) cos(y) sin(x + y) + sin(x) sin(y) cos(x + y) = 0 (Equation 2)
To solve these equations, we can use trigonometric identities:
cos(A + B) = cos(A) cos(B) - sin(A) sin(B)
sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
By applying these identities to Equation 1 and Equation 2, we get:
cos(x) sin(y) cos(x) cos(y) + sin(x) sin(y) sin(x) sin(y) = 0
sin(x) cos(y) cos(x) cos(y) + sin(x) sin(y) sin(x) sin(y) = 0
Simplifying further:
cos(x) cos(y) = -1/2 (Equation 3)
sin(x) sin(y) = 0 (Equation 4)
From Equation 4, we have two cases:
a) sin(x) = 0 or x = n*PI, where n is an integer
b) sin(y) = 0 or y = m*PI, where m is an integer
For Equation 3, we have:
cos(x) cos(y) = -1/2
By examining the unit circle, we find several solutions within the given range:
(x, y) = (π/3, π/3), (5π/3, π/3), (π/3, 5π/3), (5π/3, 5π/3)
2) Boundary Points:
Next, we examine the function at the boundary points of the region.
The boundary of the given square consists of four line segments: x = 0, x = PI, y = 0, and y = PI.
For x = 0, the function becomes:
f(0, y) = sin(0) sin(y) sin(0 + y) = 0
For x = PI, the function becomes:
f(PI, y) = sin(PI) sin(y) sin(PI + y) = 0
For y = 0, the function becomes:
f(x, 0) = sin(x) sin(0) sin(x + 0) = 0
For y = PI, the function becomes:
f(x, PI) = sin(x) sin(PI) sin(x + PI) = 0
Therefore, at the boundary points, the function has a value of 0.
3) Summary of Results:
From the critical points, we found the following solutions within the given range:
(x, y) = (π/3, π/3), (5π/3, π/3), (π/3, 5π/3), (5π/3, 5π/3)
From the boundary points, we found that the function is 0.
Now, we evaluate the function at these points to determine the absolute extreme values.
f(π/3, π/3) = sin(π/3) sin(π/3) sin(2π/3) ≈ 0.39052
f(5π/3, π/3) = sin(5π/3) sin(π/3) sin(4π/3) ≈ -0.39052
f(π/3, 5π/3) = sin(π/3) sin(5π/3) sin(4π/3) ≈ -0.39052
f(5π/3, 5π/3) = sin(5π/3) sin(5π/3) sin(10π/3) ≈ 0.39052
Therefore, the absolute extreme values of the function f(x, y) = sin(x) sin(y) sin(x + y) on the given square (0 < x < PI, 0 < y < PI) are approximately 0.39052 and -0.39052.