The minimum value of `x^(x)` is attained when x is equal to

Options

A. e
B. `e^(-1)`
C. 1
D. `e^(2)`

Answer

To find the minimum value of the function \( y = x^x \), we will follow these steps:

### Step 1: Define the function
Let \( y = x^x \).

### Step 2: Differentiate the function
To find the critical points, we need to differentiate \( y \) with respect to \( x \). We can use the property of logarithms to differentiate \( y \):
\[ \ln y = x \ln x \]
Differentiating both sides with respect to \( x \):
\[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \]
Multiplying through by \( y \):
\[ \frac{dy}{dx} = x^x (\ln x + 1) \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ x^x (\ln x + 1) = 0 \]
Since \( x^x \) is never zero for \( x > 0 \), we can focus on the term \( \ln x + 1 \):
\[ \ln x + 1 = 0 \]

### Step 4: Solve for \( x \)
Rearranging gives:
\[ \ln x = -1 \]
Exponentiating both sides:
\[ x = e^{-1} = \frac{1}{e} \]

### Step 5: Verify if it's a minimum
To confirm that this critical point is a minimum, we can check the second derivative or analyze the behavior of the first derivative around this point. However, since we are only asked for the value of \( x \) where the minimum occurs, we can conclude here.

### Final Answer
The minimum value of \( x^x \) is attained when \( x = \frac{1}{e} \).

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