Let $f:[0, \infty) \rightarrow [0, 3]$ be a function defined by
$f(x) = \begin{cases} \max \{\sin t : 0 \leq t \leq x\}, & 0 \leq x \leq \pi \\ 2 + \cos x, & x > \pi \end{cases}$
Then which of the following is true?

Given the following properties of a function $f(x)$:
$(i)$ $f(x)$ is continuous and defined for all real numbers.
$(ii)$ $f'(-5) = 0$; $f'(2)$ is not defined and $f'(4) = 0$.
$(iii)$ $(-5, 12)$ is a point on the graph of $f(x)$.
$(iv)$ $f''(2)$ is undefined,but $f''(x)$ is negative everywhere else.
$(v)$ The signs of $f'(x)$ are given by the following number line:
$f'(x)$ is positive for $x < -5$,negative for $-5 < x < 2$,positive for $2 < x < 4$,and negative for $x > 4$.
On the possible graph of $y = f(x)$,we have:

Let $f(x)=x^2+a x+b$,where $a, b \in R$. If $f(x)=0$ has all its roots imaginary,then the roots of $f(x)+f^{\prime}(x)+f^{\prime \prime}(x)=0$ are

Let $[x]$ denote the greatest integer function,and let $m$ and $n$ respectively be the numbers of the points,where the function $f(x) = [x] + |x - 2|$,$-2 < x < 3$,is not continuous and not differentiable. Then $m + n$ is equal to:

Let $f(x) = ax^2 - b|x|$,where $a$ and $b$ are constants. Then at $x = 0$,$f(x)$ has

The line which is parallel to the $x$-axis and intersects the curve $y = \sqrt{x}$ at an angle of $\frac{\pi}{4}$ is: