Explanation:

The function that we are given is sin(x)(1 - cos(x)). To find the maximum value of this function, we need to determine the critical points and analyze the behavior of the function around these points.

Finding the Critical Points:

To find the critical points, we need to take the derivative of the function and set it equal to zero.

Let's differentiate the function with respect to x:

f(x) = sin(x)(1 - cos(x))

f'(x) = cos(x)(1 - cos(x)) + sin(x)(sin(x))

Simplifying the derivative:

f'(x) = cos(x) - cos^2(x) + sin^2(x)

Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite the derivative as:

f'(x) = cos(x) - cos^2(x) + (1 - cos^2(x))

f'(x) = cos(x) - cos^2(x) + 1 - cos^2(x)

f'(x) = 2 - 2cos^2(x) - cos(x)

Setting the derivative equal to zero:

2 - 2cos^2(x) - cos(x) = 0

2cos^2(x) + cos(x) - 2 = 0

Now, we can solve this quadratic equation for cos(x). Let's denote cos(x) as t:

2t^2 + t - 2 = 0

Factoring the quadratic equation:

(2t - 1)(t + 2) = 0

This gives us two possible values for t:

t = 1/2 or t = -2

Analyzing the Behavior:

We have found the possible values for cos(x), but we need to determine which values of x correspond to these values of cos(x). We can do this by considering the range of cos(x) which is -1 to 1.

1) When cos(x) = 1/2:

Since 1/2 is within the range of cos(x), we have a valid solution.

2) When cos(x) = -2:

Since -2 is not within the range of cos(x), we don't have a valid solution for this case.

Maximum Value:

To find the maximum value of the function, we need to evaluate the function at the critical points.

1) When cos(x) = 1/2:

Plugging this value into the function:

f(x) = sin(x)(1 - cos(x))

f(x) = sin(x)(1 - 1/2)

f(x) = sin(x)(1/2)

Since the range of sin(x) is -1 to 1, the maximum value of sin(x) is 1. Therefore, the maximum value of the function sin(x)(1 - cos(x)) occurs when cos(x) = 1/2 and is equal to 1/2.

Conclusion:

The maximum value of the function sin(x)(1 - cos(x)) is 1/2. This occurs when cos(x) = 1/2.