Monotonically Increasing Functions

Q. Let $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$. Then $f$ is an increasing function in the interval :

1. $[\frac{5\pi }{8},\frac{3\pi }{4}]$
2. $[\frac{\pi }{2},\frac{5\pi }{8}]$
3. $[0,\frac{\pi }{4}]$
4. $[\frac{\pi }{4},\frac{\pi }{2}]$

Q. Let $f\left(x\right)=\frac{x^2+2}{\left[x\right]},1\le x\le 3,$ where [.] denotes greatest integer function, then incorrect statement is

1. $f\left(x\right)$ is increasing in $[1,3]$
2. least value of $f\left(x\right)$ is 3
3. $f\left(x\right)$has no greatest value
4. domain of $f^{\prime }\left(x\right)$ is $(1,3)-\left\{2\right\}$

Q. $f\left(x\right)=\{\begin{array}{c}x|x|x\le -1\\ [1+x]+[1-x]-1<x<1\\ -x|x|x\ge 1\end{array}$, then $f\left(x\right)$ is

1. an even function
2. both even as well as odd function
3. an odd function
4. neither even nor odd function

Q. Let $f:R\to R$ and $g:R\to R$ be two non-constant differentiable functions. If
$f^{\prime }\left(x\right)=\left(e^{\left(f\right(x)-g(x\left)\right)}\right)g^{\prime }\left(x\right)$ for all $x\in R$, and $f\left(1\right)=g\left(2\right)=1$, then which of the following statement(s) is (are) TRUE?

1. $f\left(2\right)<1-{\log }_{e}2$
2. $f\left(2\right)>1-{\log }_{e}2$
3. $g\left(1\right)>1-{\log }_{e}2$
4. $g\left(1\right)<1-{\log }_{e}2$

Q. What is the condition for a function y = f(x) to be a monotonically 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. $x_1>x_2\Rightarrow f\left(x_1\right)<f\left(x_2\right)$

Q. If $f:R\to R$ defined by $f\left(x\right)=\{\begin{array}{cc}x,& x<1\\ x^2,& 1\le x\le 4\\ 8\sqrt{x},& x>4\end{array}$
, then $f^{-1}\left(x\right)$ is

1. $f^{-1}\left(x\right)=\{\begin{array}{cc}x,& x<1\\ \sqrt{x},& 1\le x\le 4\\ \frac{x^2}{64},& x>4\end{array}$
2. $f^{-1}\left(x\right)=\{\begin{array}{cc}x,& x<1\\ \sqrt{x},& 1\le x\le 16\\ \frac{x^2}{64},& x>16\end{array}$
3. $f^{-1}\left(x\right)=\{\begin{array}{cc}x,& x<1\\ \sqrt{x},& 1\le x\le 8\\ \frac{x^2}{8},& x>8\end{array}$
4. $f^{-1}\left(x\right)=\{\begin{array}{cc}x,& x<1\\ \sqrt{x},& 1\le x\le 4\\ \frac{x^2}{4},& x>4\end{array}$

Q. The function $f\left(x\right)=[x{]}^2-[x^2],$ where $\left[y\right]$ is the greatest integer less than or equal to $y$, is discontinuous at

1. all integers
2. all integers except $-1$
3. all integers except $0$
4. all integers except $1$

Q. The domain of the function $f\left(x\right)=\sqrt{3+2\left[x\right]-[x{]}^2},$ where $[.]$ represents the greatest integer function is

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

Q. In the interval $x\in [0,1]$ the value of ${\cos }^{-1}\sqrt{1-x}+{\sin }^{-1}\sqrt{1-x}$ is

1. $0$
2. $\begin{array}{c}\frac{\pi }{2}\end{array}$
3. $1$
4. $\begin{array}{c}\pi \end{array}$

Q. If trace of the square matrix $A=[a_{ij}{]}_{n\times n}$ is zero, where $a_{ii}=i(i-3),$ then the order of the matrix is

1. $3\times 3$
2. $2\times 2$
3. $6\times 6$
4. $4\times 4$

Q. If $f\left(x\right)=\{\begin{array}{cc}\left[x\right],& -2\le x\le -\frac{1}{2}\\ 2x^2-1,& -\frac{1}{2}<x\le 2\end{array};$ where $[.]$ represents the greatest integer function, then the function $f(x-1)$ is discontinous at the points

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

Q. The domain of $f\left(x\right)={\sin }^{-1}\left(\frac{1}{x^2+3}\right)$ is

1. $[-\frac{\pi }{2},\frac{\pi }{2}]$
2. $(0,\frac{1}{3}]$
3. $R$
4. $[-1,1]$