Test for Monotonicity in an Interval

Q. Let $f\left(x\right)=x+\ln x-x\ln x,x\in (0,\infty )$

• Column 1 contains information about zeros of $f\left(x\right)$, $f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$.
• Column 2 contains information about the limiting behavior of $f\left(x\right)$, $f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$ at infinity.
• Column 3 contains information about increasing/decreasing nature of $f\left(x\right)$ and $f^{\prime }\left(x\right)$.

​​​​​​$\begin{array}{ccc}Column1& Column2& Column3\\ \left(I\right)f\left(x\right)=0\text{for some}x\in (1,e^2)& \left(i\right)\underset{x\to \infty }{lim}f\left(x\right)=0& \left(P\right)f\text{is increasing in}(0,1)\\ \left(II\right)f^{\prime }\left(x\right)=0\text{for some}x\in (1,e)& \left(ii\right)\underset{x\to \infty }{lim}f\left(x\right)=-\infty & \left(Q\right)f\text{is decreasing in}(e,e^2)\\ \left(III\right)f^{\prime }\left(x\right)=0\text{for some}x\in (0,1)& \left(iii\right)\underset{x\to \infty }{lim}{f}^{\prime }\left(x\right)=-\infty & \left(R\right)f^{\prime }\text{is increasing in}(0,1)\\ \left(IV\right)f^{\prime \prime }\left(x\right)=0\text{for some}x\in (1,e)& \left(iv\right)\underset{x\to \infty }{lim}{f}^{\prime \prime }\left(x\right)=0& \left(S\right)f^{\prime }\text{is decreasing in}(e,e^2)\end{array}$

Which of the following options is the only CORRECT combination?

1. $\left(II\right)\left(ii\right)\left(Q\right)$
2. $\left(III\right)\left(iii\right)\left(R\right)$
3. $\left(IV\right)\left(iv\right)\left(S\right)$
4. $\left(I\right)\left(i\right)\left(P\right)$

Q. Let $f\left(x\right)=3{\sin }^{4}x+10{\sin }^{3}x+6{\sin }^{2}x-3,$ $x\in [-\frac{\pi }{6},\frac{\pi }{2}]$.Then, $f$ is

1. decreasing in $(-\frac{\pi }{6},0)$
2. decreasing in $(0,\frac{\pi }{2})$
3. increasing in $(-\frac{\pi }{6},0)$
4. increasing in $(-\frac{\pi }{6},\frac{\pi }{2})$

Q. If $R$ is the least value of $a$ such that the function $f\left(x\right)=x^2+ax+1$ is increasing on $[1,2]$ and $S$ is the greatest value of $a$ such that the function $f\left(x\right)=x^2+ax+1$ is decreasing on $[1,2]$ then the value of $|R-S|$ is

Q. Let $f:R\to (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?

1. $x^9-f\left(x\right)$
2. $x-\underset{0}{\overset{\frac{\pi }{2}-x}{\int }}f\left(t\right)\cos tdt$
3. $e^x-\underset{0}{\overset{x}{\int }}f\left(t\right)\sin tdt$
4. $f\left(x\right)+\underset{0}{\overset{\frac{\pi }{2}}{\int }}f\left(t\right)\sin tdt$

Q. At what points in the interval [0, 2π], does the function sin 2 x attain its maximum value?

Q. Let $f:R\to R$ be a periodic function such that
$f(T+x)=1+{[1-3f(x)+3f^2(x)-f^3(x\left)\right]}^{1/3},$ where $T$ is a fixed positive number. Then period of $f\left(x\right)$ is

1. $T$
2. $3T$
3. $2T$
4. $4T$

Q.

Examine the following functions for continuity :

$f(\times )=\frac{{\times }^2-25}{\times +5},x\times not=-5$

Q.

Find the intervals on which $f\left(x\right)=x^4-8x^2$ is increasing and decreasing.

Q. The least positive values of x satisfy the equation $8^{1+\left|\cos x\right|+{\cos }^{2}x+\left|{\cos }^{3}x\right|+...\infty =4^3}$ will be (where $|$cos x$|$ $<$ 1)

1. none of these
2. 
3. 
4. 

Q.

$f\left(x\right)=1+\left[cosx\right]x,in0<x⩽\frac{\pi }{2}$

1. Is continuos in

2. Has a minimum value 0
3. Has a maximum value 2
4. Is not differentiable at x =

Q.

Examine the following functions for continuity :

$f\left(x\right)=\frac{1}{x-5},x\ne 5$

Q. Let $f\left(x\right)=x{\cos }^{-1}\left(\sin \right(-|x|\left)\right),x\in (-\frac{\pi }{2},\frac{\pi }{2})$, then which of the following is true ?

1. $f^{\prime }\left(0\right)=-\frac{\pi }{2}$
2. $f^{\prime }$ is decreasing in $(-\frac{\pi }{2},0)$ and increasing in $(0,\frac{\pi }{2})$
3. $f$ is not differentiable at $x=0$
4. $f^{\prime }$ is increasing in $(-\frac{\pi }{2},0)$ and decreasing in $(0,\frac{\pi }{2})$

Q.

Which of the following function are monotonic in the interval (0, 1) ?

1. f(x) = x²

2. f(x) = 1/x

3. f(x) = x³ - x

4. f(x) = ln(x)

Q.

Check whether the function f given by $f\left(x\right)=x^{100}+sinx-1$ strictly decreasing for the given interval.

$(0,\frac{\pi }{2})$

Q. If $f^{\prime }\left(\sin \theta \right)<0$ and $f^{\prime \prime }\left(\sin \theta \right)>0\forall \theta \in (0,\frac{\pi }{2})$ and $g\left(\theta \right)=f\left(\sin \theta \right)+f\left(\cos \theta \right),$ then

1. $g\left(\theta \right)$ is decreasing in $x\in (\frac{\pi }{4},\frac{\pi }{2})$
2. $g\left(\theta \right)$ is decreasing in $x\in (0,\frac{\pi }{4})$
   
3. $g\left(\theta \right)$ is increasing in $x\in (\frac{\pi }{4},\frac{\pi }{2})$
4. $g\left(\theta \right)$ in increasing in $x\in (0,\frac{\pi }{4})$

Q.

Check whether the function f given by $f\left(x\right)=x^{100}+sinx-1$ strictly decreasing in the given interval.

$(\frac{\pi }{2},\pi )$

Q. Let the function $f:(0,\pi )\to R$ be defined by $f\left(\theta \right)=(\sin \theta +\cos \theta {)}^2+(\sin \theta -\cos \theta {)}^4.$ Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{{\lambda }_1\pi ,\dots ,{\lambda }_r\pi \},$ where $0<{\lambda }_1<\cdots <{\lambda }_r<1.$ Then the value of ${\lambda }_1+\cdots +{\lambda }_r$ is

Q. If $f\left(x\right)=x^{11}+x^9-x^7+x^3+1$ and $f\left({\sin }^{-1}\right(\sin 8\left)\right)=\alpha ,$ where $\alpha$ is constant, then $f\left({\tan }^{-1}\right(\tan 8\left)\right)$ is equal to

1. $\alpha$
2. $\alpha -2$
3. $\alpha +2$
4. $2-\alpha$

Q.

Which of the following function are monotonic in the interval (0, 1) ?

1. $f\left(x\right)=x^2$

2. $f\left(x\right)=\frac{1}{x}$

3. $f\left(x\right)=x^3-x$

4. $f\left(x\right)=ln\left(x\right)$

Q. The function f(x)= $2x^3-9x^2+12x+4$ is decreasing in the interval(s)

1. (–∞, 1) only
2. (1, 2)
3. (–∞, 1) and (2, ∞)
4. (2, ∞) only

Q. The function f(x)= $2x^3-9x^2+12x+4$ is decreasing in the interval(s)

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

Q. Let $f\left(x\right)=\{\begin{array}{cc}2-x,& x\le 0\\ x+1,& x>0\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}x+3,& x<1\\ x^2-2x-2,& 1\le x<2\\ x-5,& x\ge 2\end{array}$. If $g\left(f\right(x\left)\right)$ is continuous at $x=0$, then the value of $g\left(f\right(0\left)\right)$ is

1. $0$
2. $-1$
3. $-3$
4. $3$

Q. Let $f\left(\theta \right)=\frac{\cot \theta }{1+\cot \theta }$ and $\alpha +\beta =\frac{5\pi }{4},$ then the value $f\left(\alpha \right)f\left(\beta \right)$ is

1. $-\frac{1}{2}$
2. $\frac{1}{2}$
3. $2$
4. none of these

Q. Let $f:R\to (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?

1. $x-\underset{0}{\overset{\frac{\pi }{2}-x}{\int }}f\left(t\right)\cos tdt$
2. $x^9-f\left(x\right)$
3. $e^x-\underset{0}{\overset{x}{\int }}f\left(t\right)\sin tdt$
4. $f\left(x\right)+\underset{0}{\overset{\frac{\pi }{2}}{\int }}f\left(t\right)\sin tdt$

Q. If $f^{\prime }\left(\sin \theta \right)<0$ and $f^{\prime \prime }\left(\sin \theta \right)>0\forall \theta \in (0,\frac{\pi }{2})$ and $g\left(\theta \right)=f\left(\sin \theta \right)+f\left(\cos \theta \right),$ then

1. $g\left(\theta \right)$ is decreasing in $x\in (\frac{\pi }{4},\frac{\pi }{2})$
2. $g\left(\theta \right)$ is decreasing in $x\in (0,\frac{\pi }{4})$
   
3. $g\left(\theta \right)$ is increasing in $x\in (\frac{\pi }{4},\frac{\pi }{2})$
4. $g\left(\theta \right)$ in increasing in $x\in (0,\frac{\pi }{4})$

Q. Consider the function
$f\left(x\right)=\{\begin{array}{cc}x\sin \frac{\pi }{x},& x>0\\ x,& x=0\end{array}$
The number of points in $(0,1),$ where the $f^{\prime }\left(x\right)$ vanishes, is

1. $0$
2. $1$
3. $2$
4. infinite

Q. The function f(x)= $2x^3-9x^2+12x+4$ is decreasing in the interval(s)

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

Q. Assertion :If $2\sin \left(\frac{\theta }{2}\right)=\sqrt{1+\sin \theta }+\sqrt{1-\sin \theta }$ then $\frac{\theta }{2}$ lies between $2n\pi +\frac{\pi }{4}$ and $2n\pi +\frac{3\pi }{4}$. Reason: If $\frac{\pi }{4}\le \theta \le \frac{3\pi }{4}$ then $\sin \frac{\theta }{2}>0$.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q.

$f\left(x\right)=1+\left[cosx\right]x,in0<x⩽\frac{\pi }{2}$

1. Has a minimum value 0
2. Has a maximum value 2

3. Is continuos in $[0,\frac{\pi }{2}]$

4. Is not differentiable at x = $\frac{\pi }{2}$

Q. The set of values of $a$ for which $a^2-a-6<0$ and $\underset{x\to \infty }{lim}{a}^x$ does not exist is/are

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