Monotonicity in an Interval

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

1. (ln28)^ln30 < (ln30)^ln28
2. (ln 2.1)^ln2.2 > (ln 2.2)^ln2.1
3. (ln 4)^ln5 < (ln5)^ln4
4. (ln30)^ln31 < (ln31)^ln30

Q. If $f\text{'}\left(x\right)=|x|-\left\{x\right\}$ where $x$ denotes the fractional part of $x$, then $f\left(x\right)$ is decreasing in

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

Q.

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

1. [0, 1)

2. [1 /2, 3 /2 ]

3. [1, 2)

4. [3 /2 , 5 /2]

Q.

Functions which are not monotonic throughout their domains could be monotonic in an interval of their domain.

1. False

2. True

Q.

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If ${\alpha }_{1}and{\beta }_1$ are real, then f(x) has vertical asymptote at $x=({\alpha }_1,{\beta }_1$).
If the equations $x^2$ + bx + c = 0 and $x^2+b_{1}x+c_1=0$ do not have real roots, then

1. f'(x) = 0 has imaginary roots

2. nothing can be said

3. f'(x) = 0 has real and distinct roots

4. f'(x) = 0 has real and equal roots

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. 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. If $f\left(x\right)=2x+{\cot }^{-1}x+\log \left(\sqrt{1+x^2}-x\right)$, then f(x)

1. increases in $\left(0,\infty \right)$
2. decreases in $\left(0,\infty \right)$
3. neither increases nor decreases in $\left(0,\infty \right)$
4. increases in $\left(-\infty ,\infty \right)$

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 $\left(0,1\right)\cup (1,e)$

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, ∞).