If $f(x) = 4x^3 - x^2 - 2x + 1$ and $g(x) = \begin{cases} \min \{f(t) : 0 \le t \le x\} & ; 0 \le x \le 1 \\ 3 - x & ; 1 < x \le 2 \end{cases}$,then the value of $g\left( \frac{1}{4} \right) + g\left( \frac{3}{4} \right) + g\left( \frac{5}{4} \right)$ is:

Let $f(x) = \lim_{n}$ ${\rightarrow \infty} \left( \frac{n^n(x+n)(x+\frac{n}{2}) \cdots (x+\frac{n}{n})}{n!(x^2+n^2)(x^2+\frac{n^2}{4}) \cdots (x^2+\frac{n^2}{n^2})} \right)^{\frac{x}{n}}$,for all $x > 0$. Then
$(A)$ $f(\frac{1}{2}) \geq f(1)$
$(B)$ $f(\frac{1}{3}) \leq f(\frac{2}{3})$
$(C)$ $f^{\prime}(2) \leq 0$
$(D)$ $\frac{f^{\prime}(3)}{f(3)} \geq \frac{f^{\prime}(2)}{f(2)}$

Which one of the following statements is $NOT \text{ } CORRECT$?

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?

The function $f(x) = \sqrt{1 - \sqrt{1 - x^2}}$

Let $f(x) = x|x|$,$g(x) = \sin x$ and $h(x) = (g \circ f)(x)$. Then