Greatest value of `(1//x)^x` is

Options

A. e
B. `e^(1//e)`
C. `(1//e)^e`
D. None of these

Answer

To find the greatest value of the function \( f(x) = \left(\frac{1}{x}\right)^x \), we can follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function in a more manageable form:
\[ f(x) = \left(\frac{1}{x}\right)^x = x^{-x} \]

### Step 2: Take the natural logarithm
To find the maximum value, we take the natural logarithm of both sides:
\[ \ln(f(x)) = \ln(x^{-x}) = -x \ln(x) \]
Let \( g(x) = -x \ln(x) \).

### Step 3: Differentiate the function
Next, we differentiate \( g(x) \) with respect to \( x \):
\[ g'(x) = -\left(\ln(x) + 1\right) \]
This is derived using the product rule and the chain rule.

### Step 4: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ -\left(\ln(x) + 1\right) = 0 \implies \ln(x) + 1 = 0 \implies \ln(x) = -1 \]
Exponentiating both sides gives:
\[ x = e^{-1} = \frac{1}{e} \]

### Step 5: Determine if it is a maximum
To confirm that this critical point is a maximum, we can check the second derivative:
\[ g''(x) = -\frac{1}{x} \]
Since \( g''(x) < 0 \) for \( x > 0 \), it indicates that \( g(x) \) is concave down at \( x = \frac{1}{e} \), confirming that we have a maximum.

### Step 6: Calculate the maximum value
Now we substitute \( x = \frac{1}{e} \) back into the original function:
\[ f\left(\frac{1}{e}\right) = \left(\frac{1}{\frac{1}{e}}\right)^{\frac{1}{e}} = e^{\frac{1}{e}} \]

Thus, the greatest value of \( \left(\frac{1}{x}\right)^x \) is:
\[ \boxed{e^{\frac{1}{e}}} \]