Strictly Increasing Functions

Q.

What is the condition for a function y = f(x) to be a strictly increasing function.

1. $x_1>x_2\Rightarrow f\left(x_1\right)>f\left(x_2\right)$

2. $x_1>x_2\Rightarrow f\left(x_1\right)\ge f\left(x_2\right)$

3. None of these

4. $x_1>x_2\Rightarrow f\left(x_1\right)=f\left(x_2\right)$

Q.

All the monotonically increasing functions are strictly increasing functions.

1. True

2. False

Q. If $m$ is the minimum value of $k$ for which the function $f\left(x\right)=x\sqrt{kx-x^2}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m,$ then the ordered pair $(m,M)$ is equal to :

1. $(4,3\sqrt{2})$
2. $(5,3\sqrt{6})$
3. $(4,3\sqrt{3})$
4. $(3,3\sqrt{3})$

Q.

The domain of the function $f\left(x\right)=\sqrt{\left(x^{14}-x^{11}+x^6-x^3+x^2+1\right)}$ is

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

2. $[0,\infty )$

3. $(-\infty ,0]$

4. $\frac{R}{\left[0,1\right]}$

Q.

Differentiate the following functions with respect to x.

$sin(x^2+5)$

Q. The number of values of $k$ for which the equation $x^2-3x+k=0$ has two distinct roots lying in the interval $(0,1)$ are

1. three
2. two
3. infinitely many
4. no value of k satisfies the requirement

Q.

$Ifx+\frac{1}{x}=5,then(x^3+\frac{1}{x^3})-5(x^2+\frac{1}{x^2})+(x+\frac{1}{x})$ is equal to

1. 5

2. -5

3. 10

4. 0

Q.

Show that the function given by $f\left(x\right)=sinx$ is
strictly increasing in $(0,\frac{\pi }{2})$

strictly decreasing in $(\frac{\pi }{2},\pi )$

neither increasing nor decreasing in $(0,\pi )$

Q.

The value of $p$ for which $6$ lie between the roots of the equation $x^2+2\left(p-3\right)x+9=0$ is

1. $\left(\frac{3}{4},\infty \right)$

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

3. $\left(\frac{-3}{4},\frac{3}{4}\right)$

4. None of these

Q.

If functions $f:A\to B$ and $g:B\to A$ satisfy $gof=I_A$, then show that f is one-one and g is onto.

Q.

Find the least value of a such that the function f given by $f\left(x\right)=x^2+ax+1$ is strictly increasing on (1, 2).

Q.

If x is real and $k=\frac{x^2-x+1}{x^2+x+1}$ then

1. 

2. 

3. 

4. 

Q.

The domain of the function $f\left(x\right)={\cos }^{-1}\left[{\log }_3\left(\frac{x}{4}\right)\right]$ is

1. $\left[4,12\right]$

2. $\left[0,3\right]$

3. $\left[\frac{4}{3},4\right]$

4. $\left[\frac{4}{3},12\right]$

Q. Prove that the function given by is increasing in R .

Q.

Prove that the function f given by $f\left(x\right)=x^2-x+1$ is neither increasing nor decreasing strictly on (-1, 1).

Q.

Find the interval in which the following functions are strictly incerasing or decreasing

$10-6x-2x^2$

Q. Let $f\left(x\right)=\frac{\sqrt{x^2+kx+1}}{x^2-k}$ is continuous for every $x\in R$ then set of values of $k$ is

Q.

Let f(x)= {x^2 ; x>=0

{ax ; x<0

Find a for which f(x) is monotonically increasing function at x=0.

Q. The complete set of values of $k$, for which the quadratic equation $x^2-kx+k+2=0$ has equal roots, consists of

1. $2+\sqrt{12}$
2. $2-\sqrt{12}$
3. $-2-\sqrt{12}$
4. $2\pm \sqrt{12}$

Q.

Find the interval in which the following function is strictly incerasing or decreasing,

$6-9x-x^2$

Q. $f\left(x\right)=\sin x-ax$ is decreasing in $R$ if

1. $a>1$
2. $a<1$
3. $a>\frac{1}{2}$
4. $a<\frac{1}{2}$

Q.

If $f\left(x\right)=\left\{\begin{array}{l}0x=0\\ x-3x>0\end{array}\right.$, then the function $f\left(x\right)$ is

1. increasing when $x>0$

2. strictly increasing when $x>0$

3. strictly increasing at $x=0$

4. not continuous at $x=0$ and so it is not increasing when $x>0$

Q. Evaluate the mentioned below:

(v)$2x^4+5x^3+7x^2-x+41,whenx=-2-\sqrt{3}i$

Q.

What is a logarithm in simple terms?

Q.

If $f\left(x\right)=\frac{sin2x+Asinx+Bcosx}{x^3}$ is countinous at x=0 then B-2A= ___

Q. The real number $k$ for which the equation $2x^3+3x+k=0$ has two dinstinct real roots in $[0,1]$ :

1. lies between $1$ and $2$
2. lies between $2$ and $3$
3. lies between $-1$ and $0$
4. does not exist

Q.

Find the interval in which the following functions are strictly incerasing or decreasing

$x^2+2x-5$

$10-6x-2x^2$

$-2x^3-9x^2-12x+1$

$6-9x-x^2$

$(x+1{)}^3(x-3{)}^3$

Q.

If $f\left(x\right)$ be a differentiable function in $\left[2,7\right]$. If $f\left(2\right)=3$ and $f\left(x\right)\le 5$ for all $x$ in $\left(2,7\right)$, then the maximum possible value of $f\left(x\right)$ at $x=7$ is

1. $7$

2. $15$

3. $28$

4. $14$

Q. Let $f\left(x\right)=4x+8\cos x-4\ln \left\{\cos x\right(1+\sin x\left)\right\}+\tan x-2\sec x-6$. If $f\left(x\right)$ is strictly increasing $\forall x\in (0,a)$ then

1. None of these
2. $a=\frac{\pi }{6}$
3. $a=\frac{\pi }{3}$
4. $a=\frac{\pi }{2}$

Q. The point of inflection of the function $f\left(x\right)=x^3+3x^2+3x+8;x\in R$ exist at which value of $x$

1. Does not exist
2. $-1$
3. $-2$
4. $1$