Monotonicity

Q.

I am sure of his success (change into a complex sentence)

Q.

In which of the following functions, Rolle’s theorem is applicable?

1. $f\left(x\right)=\left|x\right|\text{in}-2\le x\le 2$

2. $f\left(x\right)=\tan x\text{in}0\le x\le \mathrm{\pi }$

3. $f\left(x\right)=x{(x-2)}^2\text{in}0\le x\le 2$

4. $f\left(x\right)=1+{(x-2)}^{\frac{2}{3}}\text{in}1\le x\le 3$

Q.

Constant functions are Monotonically increasing as well as monotonically decreasing functions

Q. If a, b, c are in H.P., then the value of $(\frac{1}{b}+\frac{1}{c}-\frac{1}{a})(\frac{1}{c}+\frac{1}{a}-\frac{1}{b})$, is

1. None of these
2. 
3. 
4. 

Q. Let $f:R\to R$ be a continuous function such that $f\left(x\right)+f(x+1)=2,$ for all $x\in R.$ If $I_1={\int }_0^{8}f\left(x\right)dx$ and $I_2={\int }_{-1}^{3}f\left(x\right)dx$ then the value of $I_1+2I_2$ is equal to

Q.

Which of the following function is a monotonic function?

1. f(x) =1/x²

2. f(x) = {x} ; where {} is the fractional part function.

3. f(x) = x³

4. f(x) = Sin(x)

Q.

Let$\mathrm{g}\left(\mathrm{x}\right)$ be the inverse of an invertible function $\mathrm{f}\left(\mathrm{x}\right)$ which is differentiable at$\mathrm{x}=\mathrm{c}$, then $\mathrm{g}’\left(\mathrm{f}\right(\mathrm{c}\left)\right)$ equals

1. $\mathrm{f}’\left(\mathrm{c}\right)$

2. $\frac{1}{\mathrm{f}’\left(\mathrm{c}\right)}$

3. $\mathrm{f}\left(\mathrm{c}\right)$

4. None of the above

Q.

Which of the following types of functions are called monotonic functions

1. Strictly increasing

2. Monotonically increasing

3. Strictly decreasing

4. Monotonically decreasing

Q. Evaluate ${\int }_{-1}^2(e^{3x}+7x-5)dx$ as a limit of sums.

Q. Let $f:[0,1]\to 0$ (the set of all real numbers) be
a function. Suppose the function f is twice differentiable, $f\left(0\right)=f\left(1\right)=0$ and satisfies $f^{\prime \prime }\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,xϵ[0,1]$.

Q. Let f:$[0,1]\to R$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, $f\left(0\right)=f\left(1\right)=0$ and satisfies $f"\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,x\in [0,1]$
Which of the following is true for $0<x<1$?

1. $0<f\left(x\right)<\infty$
2. $-\frac{1}{2}<f\left(x\right)<\frac{1}{2}$
3. $-\frac{1}{4}<f\left(x\right)<1$
4. $-\infty <f\left(x\right)<0$

Q. Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
$h\left(x\right)$ increases as $f\left(x\right)$ decreases for all real values of $x$ if

1. $a\in (0,3)$
2. $a\in (-2,2)$
3. $a\in (3,\infty )$
4. None of these

Q. Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
$h\left(x\right)$ increases as $f\left(x\right)$ increases for all real values of $x$ if

1. $a\in (0,4)$
2. $a\in (-2,2)$
3. $a\in [3,\infty )$
4. None of these

Q.

Integrate the rational functions.
$\int \frac{2}{(1-x)(1+x^2)}dx$

Q. A function $f:R⟶R$ satisfies the equation $f(x+y)=f\left(x\right),f\left(y\right)$ for all $x,yϵR,f\left(x\right)\ne 0$ Suppose that the function is differentiable at x=0 and $f^{{}^{\prime }}\left(0\right)=2$ prove that $f^{{}^{\prime }}\left(x\right)=2f\left(x\right)$

Q. Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
If $f\left(x\right)$ is strictly increasing function, then $h\left(x\right)$ is non-monotonic function given

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

Q. Let $f:R\to R$ be a differentiable function such that $f\left(x\right)=x^2+\underset{0}{\overset{x}{\int }}{e}^{-t}f(x-t)dt.$ Then, $y=f\left(x\right)$ is

1. injective but not surjective
2. surjective but not injective
3. bijective
4. neither injective nor surjective

Q.

Identify the function from the descriptions given below.

1. Its domain is R and Range is [0, 1)

2. f(x) = x, if x $ϵ$ [0, 1)

3. f(x) = 0 if x $ϵ$ I (set of integers)

4. f(x) = 1 + x if x $ϵ$ [-1, 0)

5. f(x) is periodic with period 1

1. Greatest integer function

2. Least integer function

3. x - [x]

4. Fractional part function

5. |sin x|

Q.

Integrate the rational functions.
$\int \frac{x}{(x^2+1)(x-1)}dx.$

Q. Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
$h\left(x\right)$ increases as $f\left(x\right)$ increases for all real values of $x$ if

1. $a\in (0,4)$
2. $a\in (-2,2)$
3. $a\in [3,\infty )$
4. None of these

Q. Find all the real values of x $ϵ$ (-2 , 3) which sastify the equation
$(x^2+x+1{)}^2+1=(x^2+x+1\left)\right(x^2+x+5)$

Q. If $y=-8x-5$ and SD of $x$ is $3$, then SD of $y$ is:

1. $8$
2. $24$
3. None of these
4. $3$

Q. The function $f\left(x\right)=2x^3-9x^2+12x+29$ is monotonically decreasing function, when

1. $x<2$
2. $x>2$
3. $x>1$
4. $1<x<2$

Q. Let f be a function such that f(x+y)=f(x)+f(y) for all x and y and $f\left(x\right)=(2x^2+3x)g\left(x\right)$ for all x where g(x) is continuous and g(0)=9 then $f^{{}^{\prime }}\left(x\right)$ is equals to

1. 9
2. 27
3. 6
4. 3

Q. Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
$h\left(x\right)$ increases as $f\left(x\right)$ decreases for all real values of $x$ if

1. $a\in (0,3)$
2. $a\in (-2,2)$
3. $a\in (3,\infty )$
4. None of these

Q. Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
If $f\left(x\right)$ is strictly increasing function, then $h\left(x\right)$ is non-monotonic function given

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

Q. Let f:$[0,1]\to R$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, $f\left(0\right)=f\left(1\right)=0$ and satisfies $f"\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,x\in [0,1]$
Which of the following is true for $0<x<1$?

1. $0<f\left(x\right)<\infty$
2. $-\frac{1}{2}<f\left(x\right)<\frac{1}{2}$
3. $-\frac{1}{4}<f\left(x\right)<1$
4. $-\infty <f\left(x\right)<0$

Q. Every invertible function is
(a) monotonic function
(b) constant function
(c) identity function
(d) not necessarily monotonic function

Q. Let f:$[0,1]\to R$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, $f\left(0\right)=f\left(1\right)=0$ and satisfies $f"\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,x\in [0,1]$
Which of the following is true for $0<x<1$?

1. $0<f\left(x\right)<\infty$
2. $-\frac{1}{2}<f\left(x\right)<\frac{1}{2}$
3. $-\frac{1}{4}<f\left(x\right)<1$
4. $-\infty <f\left(x\right)<0$

Q.

Which of the following types of functions are called monotonic functions

1. Strictly increasing

2. Monotonically increasing

3. Strictly decreasing

4. Monotonically decreasing