Strictly Increasing Functions

Q. Let $\left[x\right]$ denotes the greatest integer function of $x$. If the domain of the function $\frac{1}{[x{]}^2-7[x]+12}$ is $R-[a,b),$ then the value of $a+b$ is

Q. Let $f:R\to R$ be defined as
$f\left(x\right)=\{\begin{array}{cc}-\frac{4}{3}{x}^3+2x^2+3x,& x>0\\ 3x{e}^x,& x\le 0\end{array}$
Then $f$ is increasing function in the interval:

1. $(-3,-1)$
2. $(0,2)$
3. $(-1,\frac{3}{2})$
4. $(-\frac{1}{2},2)$

Q.

All the monotonically increasing functions are strictly increasing functions.

1. True

2. False

Q. If $m$ is the minimum value of $k$ for which the function $f\left(x\right)=x\sqrt{kx-x^2}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m,$ then the ordered pair $(m,M)$ is equal to :

1. $(4,3\sqrt{2})$
2. $(5,3\sqrt{6})$
3. $(4,3\sqrt{3})$
4. $(3,3\sqrt{3})$

Q. Which among the following functions from $R$ to $R$ is not one-one function.

1. $h\left(x\right)=x^2+x$
2. $f\left(x\right)=x^3$
3. None of the above
4. $g\left(x\right)=2x+1$

Q. The maximum value of the function $f\left(x\right)=2x^3-18x^2+48x-11$ over the set $S=\{x\in R:x^2+42\le 13x\}$ is

1. $115$
2. $21$
3. $29$
4. $129$

Q. The function f(x) = |x| - |x-1| is monotonically increasing when .

1. x < 0
2. 0 < x < 1
3. x > 1
4. x < 1

Q. Let $f:R\to R$ be defined as
$f\left(x\right)=\{\begin{array}{cc}-\frac{4}{3}{x}^3+2x^2+3x,& x>0\\ 3x{e}^x,& x\le 0\end{array}$
Then $f$ is increasing function in the interval:

1. $(-3,-1)$
2. $(0,2)$
3. $(-1,\frac{3}{2})$
4. $(-\frac{1}{2},2)$

Q. The real number $k$ for which the equation $2x^3+3x+k=0$ has two dinstinct real roots in $[0,1]$ :

1. lies between $1$ and $2$
2. lies between $2$ and $3$
3. lies between $-1$ and $0$
4. does not exist

Q. If $m$ is the minimum value of $k$ for which the function $f\left(x\right)=x\sqrt{kx-x^2}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m,$ then the ordered pair $(m,M)$ is equal to :

1. $(4,3\sqrt{2})$
2. $(5,3\sqrt{6})$
3. $(4,3\sqrt{3})$
4. $(3,3\sqrt{3})$

Q.

What is the condition for a function y = f(x) to be a strictly increasing function.

1. $x_1>x_2\Rightarrow f\left(x_1\right)>f\left(x_2\right)$

2. $x_1>x_2\Rightarrow f\left(x_1\right)=f\left(x_2\right)$

3. $x_1>x_2\Rightarrow f\left(x_1\right)\ge f\left(x_2\right)$

4. None of these

Q. If $f\left(x\right)$ be an identity function in $R$ and $g\left(x\right)={\sum }_{k=1}^3{\left(f\right(x)-(2016+k\left)\right)}^{-1},$ then

1. $g\left(x\right)$ is strictly increasing in $(2018,2019)$
2. $g\left(x\right)$ has two distinct real roots
3. slope of tangent to the curve $g\left(x\right)$ at $x=f\left(2016\right)$ is $\frac{-49}{36}.$
4. $\underset{x\to -\infty }{lim}g\left(x\right)=0$

Q. Let $f\left(x\right)=4x+8\cos x-4\ln \left\{\cos x\right(1+\sin x\left)\right\}+\tan x-2\sec x-6$. If $f\left(x\right)$ is strictly increasing $\forall x\in (0,a)$ then

1. $a=\frac{\pi }{6}$
2. $a=\frac{\pi }{3}$
3. $a=\frac{\pi }{2}$
4. None of these