Consider the functions $f_{1}(x) = x$ and $f_{2}(x) = 2 + \ln x$ for $x > 0$. The graphs of these functions intersect:

At the point $x=1$,the function $f(x) = \begin{cases} x^3-1, & 1 < x < \infty \\ x-1, & -\infty < x \leq 1 \end{cases}$ is

Let $f: R \to R$ be a differentiable function such that $f(2) = 6$ and $f'(2) = \frac{1}{48}.$ Then $\lim_{x \to 2} \int_{6}^{f(x)} \frac{4t^3}{x - 2} dt$ equals

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 $f:R \to R$ be a continuous function defined by $f(x) = \frac{1}{e^x + 2e^{-x}}$.
Statement-$1$: $f(c) = \frac{1}{3}$ for some $c \in R$.
Statement-$2$: $0 < f(x) < \frac{1}{2\sqrt{2}}$ for all $x \in R$.

Let $f(x) = \begin{cases} 3x, & x < 0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{cases}$ where $[.]$ denotes the greatest integer function. If $\alpha$ and $\beta$ are the number of points where $f$ is not continuous and is not differentiable,respectively,then $\alpha + \beta$ equals.......