GATE CS 1993 Q01.2 — Calculus: Limits, Continuity, Differentiability

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows: $$f(x)=\left\{\begin{array}{lr} c_{1} e^{x}-c_{2} \log _{\mathrm{e}}\left(\frac{1}{x}\right), & \text{ if } x>0 \\ 3 & \text{ otherwise } \end{array}\right.$$ where $c_{1}, c_{2} \in \mathbb{R}$. If $f$ is continuous at $x=0$, then $c_{1}+c_{2}=$ …

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as follows: $$f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right)$$ Which of the following statements is/are true?

Consider a function $f:(0,1) \rightarrow\{0,1\}$ defined as follows. For a real number $r \in(0,1), f(r)=1$ if the second digit after the decimal point in $r$ is one of the four digits $2,3,6$ and $7$. Otherwise, $f(r)$ is equal to $0$. The number of points in $(0,1)$ at which $f$ is discontinuous is $\_\_\_\_\_\_$. (answer in integer)

Consider the given function $f(x)$. $$f(x)=\left\{\begin{array}{ll} a x+b & \text{ for } x<1 \\ x^{3}+x^{2}+1 & \text{ for } x \geq 1 \end{array}\right.$$ If the function is differentiable everywhere, the value of $b$ must be ________. (rounded off to one decimal place)

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x)=\max \left\{x, x^3\right\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers. The set of all points where $f(x)$ is NOT differentiable is

The value of the definite integral $$\int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1}\left(4 x^{2} y-z^{3}\right) \mathrm{d} z \mathrm{~d} y \mathrm{~d} x$$ is _________. (Rounded off to the nearest integer)