he minimum value of xlogex𝑥log𝑒𝑥 is

The function f(x) = xlogex can be minimized using calculus. Here are the steps:

1. Differentiate the function: The derivative of f(x) = xlogex is f'(x) = logex + 1.

2. Set the derivative equal to zero and solve for x: logex + 1 = 0 implies that logex = -1. Since the base of the natural logarithm is e, we can write this equation in exponential form as e^(-1) = x, or x = 1/e.

3. Verify that x = 1/e is a minimum: The second derivative of f(x) = xlogex is f''(x) = 1/x. Since 1/(1/e) = e > 0, we know that x = 1/e is a minimum of the function.

So, the minimum value of xlogex is achieved at x = 1/e, and the minimum value itself is (1/e)loge(1/e) = -1/e.