QuestionThe function $f(x)=(4 x-1) e^{-3 x}$ has one critical number. Find it.

Studdy Solution
Verify that $x = \frac{7}{12}$ is a critical number:
The derivative $f'(x) = e^{-3x}(7 - 12x)$ is defined for all real $x$, so the only critical number is where $f'(x) = 0$.
Since $x = \frac{7}{12}$ makes $7 - 12x = 0$, it is indeed a critical number.
The critical number of the function is $\boxed{\frac{7}{12}}$.