Given the function `f(x) = x^(2) e^(-2x), x gt 0`. Then f(x) has the maximum value equal to

Options

A. `e^(-2)`
B. `(2e)^(-1)`
C. `e^(-1)`
D. none of these

Answer

To find the maximum value of the function \( f(x) = x^2 e^{-2x} \) for \( x > 0 \), we will follow these steps:

### Step 1: Find the derivative of the function
We will use the product rule to differentiate \( f(x) \). The product rule states that if you have two functions \( u \) and \( v \), then the derivative \( (uv)' = u'v + uv' \).

Let:
- \( u = x^2 \) and \( v = e^{-2x} \)

Now, we find \( u' \) and \( v' \):
- \( u' = 2x \)
- \( v' = -2e^{-2x} \) (using the chain rule)

Now applying the product rule:
\[ f'(x) = u'v + uv' = (2x)e^{-2x} + (x^2)(-2e^{-2x}) \]
\[ f'(x) = 2xe^{-2x} - 2x^2 e^{-2x} \]
\[ f'(x) = e^{-2x}(2x - 2x^2) \]
\[ f'(x) = 2xe^{-2x}(1 - x) \]

### Step 2: Set the derivative to zero
To find the critical points, we set \( f'(x) = 0 \):
\[ 2xe^{-2x}(1 - x) = 0 \]
This gives us two factors to consider:
1. \( 2x = 0 \) which implies \( x = 0 \) (not in our domain since \( x > 0 \))
2. \( 1 - x = 0 \) which implies \( x = 1 \)

### Step 3: Determine if it's a maximum
To determine if \( x = 1 \) is a maximum, we can use the first derivative test. We check the sign of \( f'(x) \) around \( x = 1 \).

- For \( x < 1 \) (e.g., \( x = 0.5 \)):
\[ f'(0.5) = 2(0.5)e^{-1}(1 - 0.5) = e^{-1} > 0 \quad \text{(increasing)} \]

- For \( x > 1 \) (e.g., \( x = 2 \)):
\[ f'(2) = 2(2)e^{-4}(1 - 2) = -4e^{-4} < 0 \quad \text{(decreasing)} \]

Since \( f'(x) \) changes from positive to negative at \( x = 1 \), we conclude that \( x = 1 \) is a maximum.

### Step 4: Find the maximum value
Now we calculate \( f(1) \):
\[ f(1) = 1^2 e^{-2 \cdot 1} = e^{-2} \]

Thus, the maximum value of the function \( f(x) = x^2 e^{-2x} \) for \( x > 0 \) is:
\[ \text{Maximum value} = e^{-2} \]

### Summary of Steps
1. Differentiate the function using the product rule.
2. Set the derivative equal to zero to find critical points.
3. Use the first derivative test to determine if the critical point is a maximum.
4. Calculate the function value at the maximum point.