Let $f: R \rightarrow R$ be a function defined as $f(x) = \begin{cases} 3(1 - \frac{|x|}{2}) & \text{if } |x| \leq 2 \\ 0 & \text{if } |x| > 2 \end{cases}$. Let $g: R \rightarrow R$ be given by $g(x) = f(x+2) - f(x-2)$. If $n$ and $m$ denote the number of points in $R$ where $g$ is not continuous and not differentiable,respectively,then $n+m$ is equal to $....$

Let $f$ and $g$ be real-valued functions defined on the interval $(-1, 1)$ such that $g^{\prime \prime}(x)$ is continuous,$g(0) \neq 0$,$g^{\prime}(0) = 0$,$g^{\prime \prime}(0) \neq 0$,and $f(x) = g(x) \sin x$.
$STATEMENT-1$: $\lim_{x \rightarrow 0} [g(x) \cot x - g(0) \operatorname{cosec} x] = f^{\prime \prime}(0)$.
$STATEMENT-2$: $f^{\prime}(0) = g(0)$.

$\frac{1}{e^{3x}}(e^x + e^{5x}) = a_0 + a_1x + a_2x^2 + \ldots$
$\Rightarrow 2a_1 + 2^3a_3 + 2^5a_5 + \ldots$ is equal to

Let $f: R \rightarrow R$ be a function defined by $f(x) = \begin{cases} \max_{t \leq x} \{t^3 - 3t\} & x \leq 2 \\ x^2 + 2x - 6 & 2 < x < 3 \\ [x-3] + 9 & 3 \leq x \leq 5 \\ 2x + 1 & x > 5 \end{cases}$ where $[t]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I = \int_{-2}^{2} f(x) dx$. Then the ordered pair $(m, I)$ is equal to:

In the usual notation,the value of $\Delta \nabla$ is equal to

If $f(x) = \begin{cases} \int_{0}^{x} (5 + |1-t|) \, dt, & x > 2 \\ 5x + 1, & x \leq 2 \end{cases}$,then: