Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows:

$$ f(x)=\left\{\begin{array}{cc} c_1 e^x-c_2 \log _e\left(\frac{1}{x}\right), & \text { if } x>0 \\ 3, & \text { otherwise } \end{array}\right. $$

where $c_1, c_2 \in \mathbb{R}$.

If $f$ is continuous at $x=0$, then $c_1+c_2=$ $\_\_\_\_$ . (answer in integer)

The value of $x$ such that $x>1$, satisfying the equation $\int_1^x t \ln t d t=\frac{1}{4}$ is

Consider the given function $f(x)$.

$$f(x)=\left\{\begin{array}{cc} a x+b & \text { for } x<1 \\ x^3+x^2+1 & \text { for } x \geq 1 \end{array}\right.$$

If the function is differentiable everywhere, the value of $b$ must be _________ (Rounded off to one decimal place)

Let $f(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that

$f(x) = 1 - f(2 - x)$

Which one of the following options is the CORRECT value of $\int_0^2 f(x) dx$?