Number of solutions of the equation `e(k-xlnx)=1` for `kepsilon(0,(1)/(e))` are:

Explanation

To solve the equation \( e(k - x \ln x) = 1 \) for \( k \in (0, \frac{1}{e}) \), we will follow these steps:

1. **Rewrite the equation**: Start with the given equation and isolate the term involving \( x \).
\[ e(k - x \ln x) = 1 \implies k - x \ln x = \frac{1}{e} \]
Therefore, we can rewrite it as:
\[ k - \frac{1}{e} = x \ln x \]

2. **Define a function**: Let \( f(x) = x \ln x \). We need to analyze this function to find the number of solutions to the equation \( k - \frac{1}{e} = f(x) \).

3. **Find the derivative**: Compute the derivative of \( f(x) \) to determine its behavior.
\[ f'(x) = \ln x + 1 \]
Set the derivative equal to zero to find critical points:
\[ \ln x + 1 = 0 \implies \ln x = -1 \implies x = \frac{1}{e} \]

4. **Determine the nature of the critical point**: To find whether this point is a minimum or maximum, we can check the second derivative:
\[ f''(x) = \frac{1}{x} \]
Since \( f''(x) > 0 \) for \( x > 0 \), the function \( f(x) \) has a minimum at \( x = \frac{1}{e} \).

5. **Evaluate the function at the critical point**: Calculate \( f\left(\frac{1}{e}\right) \):
\[ f\left(\frac{1}{e}\right) = \frac{1}{e} \ln\left(\frac{1}{e}\right) = \frac{1}{e} \cdot (-1) = -\frac{1}{e} \]

6. **Analyze the function's behavior**: As \( x \to 0^+ \), \( f(x) \to 0 \) and as \( x \to \infty \), \( f(x) \to \infty \). The function \( f(x) \) is continuous and increasing on \( (0, \infty) \) after the minimum point.

7. **Determine the number of solutions**: The equation \( k - \frac{1}{e} = f(x) \) implies:
- For \( k \in (0, \frac{1}{e}) \), \( k - \frac{1}{e} < 0 \).
- Since \( f(x) \) has a minimum value of \(-\frac{1}{e}\) at \( x = \frac{1}{e} \) and increases to \( 0 \) as \( x \to 0^+ \) and to \( \infty \) as \( x \to \infty \), there will be exactly **two solutions** for \( k - \frac{1}{e} = f(x) \) in the interval \( (0, \frac{1}{e}) \).

Thus, the number of solutions of the equation \( e(k - x \ln x) = 1 \) for \( k \in (0, \frac{1}{e}) \) is **2**.