Introduction:
The given function is f(x) = x^2 * e^(-x). We are required to find the value of x at which the function reaches its maximum.

Analysis:
To find the maximum of the function, we need to determine the critical points by finding the derivative of the function and setting it equal to zero.

Step 1: Finding the derivative:
To find the derivative of f(x), we can use the product rule and chain rule. The derivative of f(x) is given by:
f'(x) = 2x * e^(-x) - x^2 * e^(-x)

Step 2: Setting the derivative equal to zero:
To find the critical points, we set the derivative equal to zero and solve for x:
2x * e^(-x) - x^2 * e^(-x) = 0

Step 3: Simplifying the equation:
We can factor out e^(-x) from the equation:
e^(-x) * (2x - x^2) = 0

Step 4: Solving for x:
Setting each factor equal to zero gives us two possible solutions:
1) e^(-x) = 0 (which is not possible)
2) 2x - x^2 = 0

Solving the quadratic equation, we get:
x(2 - x) = 0
x = 0 or x = 2

Conclusion:
We have found two critical points, x = 0 and x = 2. To determine which one corresponds to the maximum, we can analyze the behavior of the function.

Step 5: Analyzing the function:
We can observe that as x approaches positive infinity, the exponential term e^(-x) approaches zero. Therefore, the function approaches zero as x increases. Similarly, as x approaches negative infinity, the exponential term e^(-x) approaches infinity, making the function approach infinity as x decreases.

Step 6: Comparing the critical points:
Considering the behavior of the function, we can conclude that x = 0 corresponds to a minimum point, and x = 2 corresponds to a maximum point. At x = 2, the function reaches its maximum.

Final Answer:
The maximum of the function f(x) = x^2 * e^(-x) occurs when x is equal to 2.