Final answer:
To find the absolute minimum value of f(x) = sin(x) cos(x) on the interval (0, 2π), find its critical points by setting the derivative equal to zero. The absolute minimum value of f(x) on the interval (0, 2π) is -1/2.
Step-by-step explanation:
The function f(x) = sin(x) cos(x) is a product of two trigonometric functions. To find the absolute minimum value of f(x) on the interval (0, 2π), we need to find the critical points of f(x) and check their values. To find the critical points, we take the derivative of f(x) and set it equal to zero:
f'(x) = (cos²(x) - sin²(x)) = 0
Using trigonometric identities, we can simplify the equation to:
2cos²(x) - 1 = 0
cos²(x) = 1/2
cos(x) = ±√(1/2)
These critical points occur at x = π/4 and x = 3π/4 in the interval (0, 2π). Plugging the values of x into f(x), we get f(π/4) = 1/2 and f(3π/4) = -1/2. Therefore, the absolute minimum value of f(x) on the interval (0, 2π) is -1/2.