QuestionFind the local minimum value of the function $f(x) = x \ln(2x)$.

Studdy Solution
Calculate the local minimum value of the function at $x = \frac{1}{2e}$:
$$f\left(\frac{1}{2e}\right) = \frac{1}{2e} \ln\left(2 \cdot \frac{1}{2e}\right)$$
$$= \frac{1}{2e} \ln\left(\frac{1}{e}\right)$$
$$= \frac{1}{2e} \cdot (-1)$$
$$= -\frac{1}{2e}$$
The local minimum value of the function is $\boxed{-\frac{1}{2e}}$.