Inequalities

Q.

Let $f:\left(-1,\infty \right)\to \mathrm{ℝ}$ be defined by $f\left(0\right)=1$ and $f\left(x\right)=\frac{1}{x}{\log }_e\left(1+x\right),x\ne 0$. Then the function $f$

1. increases in $\left(-1,\infty \right)$

2. decreases in $\left(-1,0\right)$ and increases in $\left(0,\infty \right)$

3. increases in $\left(-1,0\right)$ and decreases in $\left(0,\infty \right)$

4. decreases in $\left(-1,\infty \right)$

Q. Let $f\left(x\right)=\ln (1+x)-x,x>0.$ Then which of the following is correct

1. $f\left(x\right)>0$
2. $f\left(x\right)\ge 0$
3. $f\left(x\right)<0$
4. $f\left(x\right)\le 0$

Q. For $x>0,$ which of the following is correct

1. $\log (1+x)\le \frac{x}{1+x}$
2. $\log (1+x)<\frac{x}{1+x}$
3. $\log (1+x)\ge \frac{x}{1+x}$
4. $\log (1+x)>\frac{x}{1+x}$

Q. For $0<x_1<x_2<1,$ which of the following is correct

1. $\frac{\sin x_2}{\sin x_1}\ge \frac{\ln x_2}{\ln x_1}$
2. $\frac{\sin x_2}{\sin x_1}>\frac{\ln x_2}{\ln x_1}$
3. $\frac{\sin x_2}{\sin x_1}<\frac{\ln x_2}{\ln x_1}$
4. $\frac{\sin x_2}{\sin x_1}\le \frac{\ln x_2}{\ln x_1}$

Q. Which of the following is correct for $0<x_1<x_2<\frac{\pi }{2}$

1. $x_1^2\sin x_1-x_2^2\sin x_2>x_2\cos x_2-x_1\cos x_1$
2. $x_1^2\sin x_1-x_2^2\sin x_2<x_2\cos x_2-x_1\cos x_1$
3. $x_1^2\sin x_1-x_2^2\sin x_2\le x_2\cos x_2-x_1\cos x_1$
4. $x_1^2\sin x_1-x_2^2\sin x_2\ge x_2\cos x_2-x_1\cos x_1$

Q. If a function $f\left(x\right)=\underset{0}{\overset{x}{\int }}\text{sgn}\left(t\right)(t^2-7t+6)dt$ is defined in $x\in (0,\infty ),$ then which of the following is/are correct?

1. $f\left(x\right)$ has a local maxima at $x=1$
2. $f\left(x\right)$ is increasing on $(0,1)$
3. there exist $c\in (0,2)$ such that $f^{\prime }\left(c\right)=0$
4. $f$ is decreasing on $(0,1)$

Q. The point which lies in the half plane $3x-2y-2\ge 0$ is

[1 mark]

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

Q. The point which lies in the half plane $3x-2y-2\ge 0$ is

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

Q. For $x\ge 0,$ the minimum value of $f\left(x\right)=\ln (1+x)-x+\frac{x^2}{2}$ is

Q. If a continuous function $e^{-x}f\left(x\right)$ has only one stationary point and attains its maximum in $[1,3]$ at $x=2,$ then which of the following(s) is/are correct.

1. $f\left(x\right)<f^{\prime }\left(x\right)$ for $1<x<2$
2. $f\left(x\right)<f^{\prime }\left(x\right)$ for $2<x<3$
3. $f\left(x\right)>f^{\prime }\left(x\right)$ for $1<x<2$
4. $f\left(x\right)>f^{\prime }\left(x\right)$ for $2<x<3$

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

Q. If function $f\left(x\right)=\lambda \cos 2x+2\sin x$ has only one extremum point in $[0,\pi ],$ then the value of $\lambda$ can not be

1. $-\frac{1}{4}$
2. $\frac{2}{5}$
3. $\frac{3}{7}$
4. $\frac{7}{8}$

Q. Let a function $f\left(x\right)=x^2-4|x|,$ then which of the following is correct

1. $f\left(x\right)$ is strictly increasing in $(-\infty ,-2)$
2. $f\left(x\right)$ is strictly increasing in $(-1,1)$
3. $f\left(x\right)$ is strictly decreasing in $(-2,2)$
4. $f\left(x\right)$ is strictly decreasing in $(1,2)$

Q. Which of the following is true in the interval of $x\in (0,1)$

1. ${\sin }^{-1}x<{\tan }^{-1}x$
2. ${\sin }^{-1}x\le {\tan }^{-1}x$
3. ${\sin }^{-1}x>{\tan }^{-1}x$
4. ${\sin }^{-1}x\ge {\tan }^{-1}x$