If $f(x)$ is a function such that $f^{\prime}(x)=(x-1)^{2}(4-x),$ then

The minimum value of the function $f(x) = x \log x$ is

If $(2, a)$ and $(b, 19)$ are two stationary points of the curve $y=2x^3-15x^2+36x+c$,then $a+b+c=$

The maximum value of $f(x) = \sin (x)$ in the interval $[-\pi / 2, \pi / 2]$ is

The number of distinct real roots of the equation $3x^{4} + 4x^{3} - 12x^{2} + 4 = 0$ 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 $n$