**Maximum value of f(x) = sin(x) cos(x)**

To find the maximum value of the given function f(x) = sin(x) cos(x), we can use the properties of trigonometric functions along with calculus techniques. Let's break down the solution into steps:

1. **Identifying the Periodicity**: The function f(x) = sin(x) cos(x) can be rewritten using the identity sin(2x) = 2sin(x)cos(x). Therefore, we can rewrite the function as f(x) = (1/2)sin(2x). Now, we can see that the function has a periodicity of π, which means it repeats itself every π units.

2. **Finding the Critical Points**: To find the critical points, we need to calculate the derivative of f(x) and set it equal to zero. Differentiating f(x) = (1/2)sin(2x) with respect to x, we get f'(x) = cos(2x). Setting f'(x) = 0, we find the critical points as cos(2x) = 0. Solving this equation, we get 2x = π/2 + nπ/2, where n is an integer. This implies x = π/4 + nπ/4 for all integers n.

3. **Evaluating the Function at Critical Points**: Now, we need to evaluate the function f(x) = (1/2)sin(2x) at the critical points x = π/4 + nπ/4. Plugging in these values, we get f(x) = (1/2)sin(2(π/4 + nπ/4)) = (1/2)sin((π/2) + n(π/2)) = (1/2)sin((2n + 1)(π/2)). Since sin((2n + 1)(π/2)) takes values between -1 and 1, f(x) will also be between -1/2 and 1/2.

4. **Finding the Maximum Value**: From the above step, we can conclude that the maximum value of f(x) = sin(x) cos(x) is 1/2. This maximum value occurs when sin((2n + 1)(π/2)) = 1, which happens when (2n + 1)(π/2) = π/2 + 2mπ, where m is an integer. Simplifying, we get n = m. Therefore, the maximum value occurs at x = π/4 + nπ/4, where n is an integer.

In conclusion, the maximum value of the function f(x) = sin(x) cos(x) is 1/2 and occurs at x = π/4 + nπ/4, where n is an integer.

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