The function f(x) = x log x has maximum value 1/e.

The function f(x) has maximum value -1/e.

The function f(x) has minimum value 1/e.

The function f(x) has minimum value -1/e.

The function \( f(x) = x \log x \) is an interesting mathematical expression that can be analyzed to find its maximum and minimum values. To clarify the statements regarding its values:

Maximum and Minimum Values

For the function \( f(x) = x \log x \):

• Maximum Value: The function achieves its maximum value at \( x = \frac{1}{e} \), which is approximately 0.3679. At this point, \( f\left(\frac{1}{e}\right) = \frac{1}{e} \log\left(\frac{1}{e}\right) = -\frac{1}{e} \).
• Minimum Value: The function does not have a minimum value in the traditional sense, as it approaches negative infinity as \( x \) approaches 0 from the right.

Summary of Key Points

In summary:

• The maximum value of \( f(x) \) is \( -\frac{1}{e} \) at \( x = \frac{1}{e} \).
• The function does not have a defined minimum value.

This analysis shows how \( f(x) = x \log x \) behaves, particularly around its critical points.