Monotonicity in an Interval

Q. Which of the following is/are true, For $f\left(x\right)=\frac{ln\left(lnx\right)}{lnx}$

1. $(ln2.1{)}^{ln2.2}>(ln2.2{)}^{ln2.1}$
2. $(ln4{)}^{ln5}<(ln5{)}^{ln4}$
3. $(ln30{)}^{ln31}<(ln31{)}^{ln30}$
4. $(ln28{)}^{ln30}<(ln30{)}^{ln28}$

Q.

Interval in which {x} is monotonically increasing function, where { } is fractional part function -

1. $[0,1)$

2. $[\frac{1}{2},\frac{3}{2}]$

3. $[1,2)$

4. $[\frac{3}{2},\frac{5}{2}]$

Q. If function $f\left(x\right)=\{\begin{array}{cc}1,& x=1\\ a^2-3a+x^2,& x>1\end{array}$ has a local maximum at $x=1,$ then the set of values of $a$ is

1. $(-\infty ,3)$
2. $(3,\infty )$
3. $(-\infty ,0)$
4. $(0,3)$

Q. The function $f\left(x\right)=4{\sin }^{3}x–6{\sin }^{2}x+12\mathrm{sinx}+100$ is strictly

1. decreasing in $(\frac{\pi }{2},\frac{3\pi }{2})$
2. decreasing in $(\frac{\pi }{2},\pi )$
3. increasing in $(\frac{-\pi }{2},\frac{\pi }{2})$
4. increasing in $(\frac{\pi }{2},\frac{3\pi }{2})$

Q.

Interval in which {x} is monotonically increasing function, where { } is fractional part function -

1. $[0,1)$

2. $[\frac{1}{2},\frac{3}{2}]$

3. $[1,2)$

4. $[\frac{3}{2},\frac{5}{2}]$

Q. Which of the following is (are) correct about the function $y=x^4-\frac{4x^3}{3}$ ?

1. f is decreasing in (– ∞, 1]
2. f is strictly increasing in (– ∞, 1]
3. f is increasing in [1, ∞).
4. f is decreasing in (1, ∞).

Q. The function $f\left(x\right)=\frac{2x^2-1}{x^4},x>0$, decreases in the interval

1. $[1,\infty )$
2. $\left(1,\infty \right)$
3. $\left(0,\infty \right)$
4. $[0,\infty )$

Q. Which of the following is/are true, For $f\left(x\right)=\frac{ln\left(lnx\right)}{lnx}$

1. $(ln2.1{)}^{ln2.2}>(ln2.2{)}^{ln2.1}$
2. $(ln4{)}^{ln5}<(ln5{)}^{ln4}$
3. $(ln30{)}^{ln31}<(ln31{)}^{ln30}$
4. $(ln28{)}^{ln30}<(ln30{)}^{ln28}$

Q. Which of the following is (are) correct about the function $f\left(x\right)=\frac{x}{\log x}$

1. f(x) is increasing on $\left(e,\infty \right)$
2. f(x) is decreasing on $\left(0,\infty \right)$
3. f(x) is increasing on $\left(0,1\right)\cup (1,e)$
4. f(x) is decreasing on $\left(0,1\right)\cup (1,e)$