The function f defined by $f(x)=x^{1/x}$ has a maximum at:

Question

Accepted Answer

To determine where the function $ f(x) = x^{1/x} $ has a maximum, we need to analyze the behavior of the function. The given correct answer is Option 2: $ e $.

1. **Explanation for Option 2 (Correct Answer):**

The function $ f(x) = x^{1/x} $ can be rewritten using the natural exponential function as $ f(x) = e^{\frac{\ln x}{x}} $. To find the maximum, we need to find the critical points by taking the derivative of $ f(x) $ and setting it to zero.

First, consider the natural logarithm of the function:
   $$
   \ln(f(x)) = \frac{\ln x}{x}
   $$
   Differentiate this with respect to $ x $ using the quotient rule:
   $$
   \frac{d}{dx}\left(\frac{\ln x}{x}\right) = \frac{1 - \ln x}{x^2}
   $$
   Set the derivative equal to zero to find critical points:
   $$
   1 - \ln x = 0 \quad \Rightarrow \quad \ln x = 1 \quad \Rightarrow \quad x = e
   $$
   This shows that $ x = e $ is a critical point. To confirm it's a maximum, we check the second derivative or analyze the behavior around $ x = e $. The second derivative test or analyzing the sign change of the first derivative around $ x = e $ confirms that this is indeed a maximum.

2. **Explanation for why other options are incorrect:**

- **Option 1: $ \log_{e}2 $**

$ \log_{e}2 $ is approximately 0.693, which is less than 1. At $ x = \log_{e}2 $, the derivative $ \frac{1 - \ln x}{x^2} $ does not equal zero, indicating it is not a critical point. Therefore, it cannot be the location of a maximum.

- **Option 3: 2**

At $ x = 2 $, the derivative $ \frac{1 - \ln x}{x^2} $ is negative, indicating that the function is decreasing at this point. Thus, $ x = 2 $ is not a maximum.

- **Option 4: 1**

At $ x = 1 $, the function $ f(x) = 1^{1/1} = 1 $, and the derivative $ \frac{1 - \ln x}{x^2} $ is positive, indicating that the function is increasing at this point. Therefore, $ x = 1 $ is not a maximum.

In conclusion, the function $ f(x) = x^{1/x} $ reaches its maximum value at $ x = e $, which is why Option 2 is the correct answer.