Find the local minimum value of the function $f$ given by $f(x) = 3 + |x|, x \in R$.

In the interval $(0, 1)$,the maximum value of the function $f(x) = |x \ln x|$ is:

The difference between the greatest and least values of the function $f(x) = -x + \sin 2x$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$ is

Let $f(x) = \frac{\sin \pi x}{x^2}, x > 0$. Let $x_1 < x_2 < x_3 < \ldots < x_n < \ldots$ be all the points of local maximum of $f(x)$ and $y_1 < y_2 < y_3 < \ldots < y_n < \ldots$ be all the points of local minimum of $f(x)$. Which of the following statements are true?
$(1)$ $|x_n - y_n| > 1$ for every $n$
$(2)$ $x_1 < y_1$
$(3)$ $x_n \in (2n, 2n + \frac{1}{2})$ for every $n$
$(4)$ $x_{n+1} - x_n > 2$ for every $n$

The minimum value of $e^{(2x^2 - 2x - 1)\sin^2 x}$ is .......

The necessary condition for a function to have a maximum or minimum value is: