Let $x_0$ be the real number such that $e^{x_0} + x_0 = 0$. For a given real number $\alpha$, define
$$g(x) = \frac{3x e^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)}$$
for all real numbers $x$.
Then which one of the following statements is TRUE?
Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by
$f(x)=\left\{\begin{array}{cc}2-2 x^2-x^2 \sin \frac{1}{x} & \text { if } x \neq 0, \\ 2 & \text { if } x=0 .\end{array}\right.$
Then which one of the following statements is TRUE?
Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let [ x ] denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List–I | List–II |
|---|---|
| (P) The minimum value of $n$ for which the function $$ f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right] $$ is continuous on the interval $[1,2]$, is | (1) 8 |
| (Q) The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right)$, $x \in \mathbb{R}$, is an increasing function on $\mathbb{R}$, is | (2) 9 |
| (R) The smallest natural number $n$ which is greater than 5 , such that $x=3$ is a point of local minima of $$ h(x)=\left(x^2-9\right)^n\left(x^2+2 x+3\right) $$ is | (3) 5 |
| (S) Number of $x_0 \in \mathbb{R}$ such that
$$ l(x)=\sum\limits_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right) $$ $x \in \mathbb{R}$, is NOT differentiable at $x_0$, is |
(4) 6 |
| (5) 10 |
