Find the maximum and minimum values of $f,$ if any,of the function given by $f(x)=|x|, x \in R.$

Find all points of local maxima and local minima of the function $f$ given by $f(x) = x^3 - 3x + 3$.

Maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 27$ is

The function $f(x) = \sin^p x \cos^q x$ has a maximum at:

The radius of the cylinder of maximum volume,which can be inscribed in a sphere of radius $R$ 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 $n$