If $f(x) = \begin{cases} ax^2 - bx + 2, & x < 3 \\ bx^2 - 3, & x \geq 3 \end{cases}$ is differentiable at every $x \in R$,then the area (in sq units) of the triangle formed by the line $\frac{x}{a} + \frac{y}{b} = 1$ with the coordinate axes is

Which of the following is not true?

Let $f: R \rightarrow R$ be a differentiable function such that $f(0)=0$,$f(\frac{\pi}{2})=3$ and $f^{\prime}(0)=1$. If $g(x)=\int_x^{\pi / 2} [f^{\prime}(t) \operatorname{cosec} t - f(t) \operatorname{cosec} t \cot t] dt$ for $x \in (0, \frac{\pi}{2}]$,then $\lim _{x \rightarrow 0} g(x)=$

Two functions $f$ and $g$ have first and second derivatives at $x = 0$ and satisfy the relations: $f(0) = \frac{2}{g(0)}$,$f'(0) = 2g'(0) = 4g(0)$,$g''(0) = 5f''(0) = 6f(0) = 3$. Then:

Let $f:\left[-\frac{1}{2}, 2\right] \rightarrow R$ and $g:\left[-\frac{1}{2}, 2\right] \rightarrow R$ be functions defined by $f(x)=\left[x^2-3\right]$ and $g(x)=|x| f(x)+|4 x-7| f(x)$,where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in R$. Then
$(A)$ $f$ is discontinuous exactly at three points in $\left[-\frac{1}{2}, 2\right]$
$(B)$ $f$ is discontinuous exactly at four points in $\left[-\frac{1}{2}, 2\right]$
$(C)$ $g$ is $NOT$ differentiable exactly at four points in $\left(-\frac{1}{2}, 2\right)$
$(D)$ $g$ is $NOT$ differentiable exactly at five points in $\left(-\frac{1}{2}, 2\right)$

Which of the following statements is true for the function $f(x) = \begin{cases} \sqrt{x} & x \ge 1 \\ x^3 & 0 \le x < 1 \\ \frac{x^3}{3} - 4x & x < 0 \end{cases}$