Critical Point

Q.

If the functions are defined as $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=\sqrt{1-x}$ then what is the common domain of the following functions: $f+g,f-g,\frac{f}{g},\frac{g}{f}$?

1. $0<x\le 1$

2. $0\le x<1$

3. $0\le x\le 1$

4. $0<x<1$

Q.

Critical points are the points where f'(x) is zero or it doesn't exist

1. True

2. False

Q. The set of values of $a$ for which the function $f:R\to R$ given by $f\left(x\right)=x^3+(a+2)x^2+3ax+5$ is one-one, is

1. $(-\infty ,1)\cup (4,\infty )$
2. $(-\infty ,4)$
3. $(1,\infty )$
4. $[1,4]$

Q. If the function $f$ given by
$f\left(x\right)=x^3-3(a-2)x^2+3ax+7$, for some $a\in R$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation,
$\frac{f\left(x\right)-14}{(x-1{)}^2}=0(x\ne 1)$ is :

1. $-7$
2. $5$
3. $6$
4. $7$

Q. The minimum value of $\alpha$ for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=\alpha$ has at least one solution in $(0,\frac{\pi }{2})$ is

Q. If the function $f$ given by
$f\left(x\right)=x^3-3(a-2)x^2+3ax+7$, for some $a\in R$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation,
$\frac{f\left(x\right)-14}{(x-1{)}^2}=0(x\ne 1)$ is :

1. $-7$
2. $5$
3. $6$
4. $7$

Q. If $f\left(x\right)=\frac{ax+b}{(x-1)(x-4)}$ has local maximum point at $(2,-1),$ then $(a+b)$ is equal to