Monotonically Decreasing Functions

Q. Let $f,g:N\to N$ such that $f(n+1)=f\left(n\right)+f\left(1\right),\forall n\in N$ and $g$ be any arbitrary function. Which of the following statements is $\text{NOT}$ true $?$

1. $f$ is one -one
2. If $g$ is onto, then $fog$ is one-one
3. If $f$ is onto, then $f\left(n\right)=n\forall n\in N$
4. If $fog$ is one-one, then $g$ is one-one

Q. Let $\phi \left(x\right)=f\left(x\right)+f(1-x)$ and $f"\left(x\right)<0$ in $[0,1],$ then

1. $\phi$ is monotonic increasing in $[0,\frac{1}{2}]$ and monotonic decreasing in $[\frac{1}{2},1]$
2. $\phi$ is monotonic increasing in $[\frac{1}{2},1]$ and monotonic decreasing in $[0,\frac{1}{2}]$
3. $\phi$ is neither increasing nor decreasing in any sub interval of $[0,1]$
4. $\phi$ is increasing in $[0,1]$

Q. Let $F:R\to R$ be a thrice differentiable function. Supose that $F\left(1\right)=0,F\left(3\right)=–4$ and $F^{\prime }\left(x\right)<0$ for all $x\in (1/2,3)$. Let $f\left(x\right)=xF\left(x\right)$ for all $x\in R$.

The correct statement(s) is(are)

1. $f^{\prime }\left(1\right)<0$
2. $f\left(2\right)<0$
3. $f^{\prime }\left(x\right)\ne 0$ for any $x\in (1,3)$
4. $f^{\prime }\left(x\right)=0$ for some $x\in (1,3)$

Q.

Let $f\left(x\right)=\left[x\right]\sin \left(\frac{\pi }{\left[x+1\right]}\right)$, where $\left[x\right]$ denotes the greatest integer function. The domain of $f$ is

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

2. $[0,\infty )$

3. $(-\infty ,-1)\cup [0,\infty )$

4. None of these

Q. Let $f\left(x\right)$ be a non-negative function. If $f^{\prime }\left(x\right)\cos x\le f\left(x\right)\sin x,\forall x\ge 0,$ then value of $f\left(\frac{5\pi }{3}\right)$ is

1. $0$
2. $\frac{\sqrt{3}}{2}$
3. $\frac{1}{2}$
4. $1$

Q.

The function $f$ defined by $f\left(x\right)=x^3-6x^2-36x+7$ is increasing if

1. $x>2$ and also $x>6$.

2. $x>2$ and also $x<6$.

3. $x>-2$ and also $x<6$.

4. $x<-2$ and also $x>6$.

Q.

The function $f\left(x\right)=1-x^3$:

1. increases everwhere

2. decreases in $(0,\infty )$

3. increases in $(0,\infty )$

4. None of the above

Q. Let f(x) be a function such that $f\left(x\right)={\log }_{1/3}\left({\log }_3\left(\sin x+a\right)\right]$. If f(x) is decreasing for all real values of x, then .

1. $a\in \left(1,4\right)$
2. $a\in \left(4,\infty \right)$
3. $a\in \left(2,3\right)$
4. $a\in \left(2,\infty \right)$

Q. If $f\left(x\right)=x^3+4x^2+\mathrm{kx}+1$ is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then .

1. k = 4
2. k = 2
3. k = -1
4. k has no real value

Q. If f(x) = $a{x}^2$ + bx + c, then the value of a and b for which the identify f(x+1) - f(x) = 8x + 3 is satisfied, are

1. a =1 and b = 4
2. a = 1 and b = - 4
3. a = 4 and b = 1
4. a = 4 and b = -1

Q. 10. The function f(x)=x+1/x("x not equal to zero") is a non-increasing function in the interval .

Q. Let $f\left(x\right)=\frac{x(\sin x+\tan x)}{\left[\frac{x+\pi }{\pi }\right]-\frac{1}{2}}$, then which among the following option is incorrect
(where [.] denotes the greatest integer function)

1. $f\left(x\right)$ is an odd function when $x\ne n\pi$
2. $f\left(x\right)$ is an even function when $x=n\pi$
3. $f\left(x\right)$ is an odd function when $x=n\pi$
4. $f\left(x\right)$ is an even function when $x\ne n\pi$

Q. Which of the following functions are decreasing on $(0,\frac{\pi }{2})$ ?

1. $\tan x$
2. $\cos 3x$
3. $\cos 2x$
4. $\cos x$

Q.

Monotonically decreasing functions are also called as non - increasing functions.

1. True

2. False

Q. Let $f:R\to R$ be a differentiable function for all values of x and has a property that f(x) and f (x) have opposite signs for all values of x. Then,

1. f(x) is an increasing function.
2. f(x) is a decreasing function.
3. $f^2\left(x\right)$ is a decreasing function.
4. |f(x)| is an increasing function.

Q. Let $f\left(x\right)=\cos x\cdot \sin 2x$ and $x\in [0,\frac{\pi }{2}],$ then

1. $f\left(x\right)$ has global minimum at $x=0$
2. $f\left(x\right)$ has global maximum at $x=\frac{\pi }{2}$
3. Global maximum value of $f\left(x\right)$ is $1$
4. Global maximum value of $f\left(x\right)$ is $\frac{4}{3\sqrt{3}}$

Q.

if the roots of the equation x² -2mx +m² - 1 =0 lie in the interval (-2, 4) then,

a) m ~ (-1, 3) b) m~(1, 5)

c) m~(-3, 1) d) m~(-1.5)

where ~ stand for "belongs to".

Q. Show that the function given by $f\left(x\right)=\sin x$ is
(a) increasing in $(0,\frac{\pi }{2})$
(b) strictly decreasing $(\frac{\pi }{2},\pi )$
(c)neither increasing nor decreasing in $(0,\pi )$

Q. Let $f:R\to R$ be an invertible and a differentiable function defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2+ax-6,& x\le 2\\ \alpha x^2+\beta ,& x>2\end{array}$ where $a,\beta \in Z,\alpha \in R$ and $a\ge -5.$ If $f^{-1}$ denotes the inverse function of $f,$ then

1. $a+4\alpha +\beta =-17$
2. $f^{-1}(-12)=3$
3. $\left(f^{-1}{)}^{\prime }\right(0)=-\frac{1}{7}$
4. $\left(f^{-1}\right)(-12)=2$

Q.

Which of the following functions are monotonically decreasing functions ?

1. f(x) = 2x +3

2. f(x) = 5 - 8x

3. f(x) = 3

4. None of these

Q. Let $F:R\to R$ be a thrice differentiable function. Supose that $F\left(1\right)=0,F\left(3\right)=–4$ and $F^{\prime }\left(x\right)<0$ for all $x\in (1/2,3).$ Let $f\left(x\right)=xF\left(x\right)$ for all $x\in R$.

The correct statement(s) is(are)

1. $f^{\prime }\left(1\right)<0$
2. $f\left(2\right)<0$
3. $f^{\prime }\left(x\right)\ne 0$ for any $x\in (1,3)$
4. $f^{\prime }\left(x\right)=0$ for some $x\in (1,3)$

Q. the domain of f(x)= cube root(2x-1/x^2-10x-11)

Q. If X and Y are two non- empty sets where f;$X\to Y$ is function is defined such that$\left\{f\right(c)=y:c\in X\}$ for $c\subseteq X$, $y\subseteq Y$ and $\left\{f^{-1}\right(D)=x:f(x)\in D\}$ for $D\subseteq Y$ for any $x\subseteq X$ , then

1. f^-1 (f(A)) = A
2. f^-1(f(A))=A only if f(x) = Y
3. f(f^-1 (B)) = B only if B ⊆ f(x)
4. f(f^-1(B)) = B

Q. For a function $f,f(a+b-x)=f\left(x\right)$, and ${\int }_a^{b}xf\left(x\right)dx=k{\int }_a^{b}f\left(x\right)dx$, then value of $k$ is

1. $\frac{1}{2}(a+b)$
2. $\frac{1}{2}(a-b)$
3. $\frac{1}{2}(a^2+b^2)$
4. $\frac{1}{2}(a^2-b^2)$

Q. Let $F:R\to R$ be a thrice differentiable function. Supose that $F\left(1\right)=0,F\left(3\right)=–4$ and $F^{\prime }\left(x\right)<0$ for all $x\in (1/2,3).$ Let $f\left(x\right)=xF\left(x\right)$ for all $x\in R$.

The correct statement(s) is(are)

1. $f^{\prime }\left(1\right)<0$
2. $f\left(2\right)<0$
3. $f^{\prime }\left(x\right)\ne 0$ for any $x\in (1,3)$
4. $f^{\prime }\left(x\right)=0$ for some $x\in (1,3)$

Q. If $\frac{d}{dx}f\left(x\right)=g\left(x\right)$ for $a\le x\le b$, then $\stackrel{b}{\underset{a}{\int }}f\left(x\right)g\left(x\right)dx$ equals to:

1. $f\left(b\right)-f\left(a\right)$
2. $\frac{\left[g\right(b){]}^2-[g\left(a\right){]}^2}{2}$
3. $g\left(b\right)-g\left(a\right)$
4. $\frac{\left[f\right(b{)}^2]-[f\left(a\right){]}^2}{2}$

Q. Let $f:R\to R$ be a function such that $f\left(f\right(x\left)\right)=x^2-x+1$ for all $x$ and the value of ${\tan }^{-1}\left(f\right(1\left)\right)+{\sec }^{-1}\left(f\right(1\left)\right)+{\cos }^{-1}\left(f\right(0\left)\right)+{\cot }^{-1}\left(f\right(0\left)\right)$ is equal to $\frac{a\pi }{b}$ where $a$ and $b$ are co-prime. Then the value of $a+b$ is

Q. Let $f\left(x\right)=a{x}^2+bx+c$. lf $f(-1)<1,f\left(1\right)>-1,f\left(3\right)<-4$ and $a\ne 0$, then

1. $a<-\frac{1}{8}$
2. $a>0$
3. $a>1$
4. $-\frac{1}{8}<a<0$

Q. $\underset{0}{\overset{1}{\int }}{\cot }^{-1}\left(1-x+x^2\right)dx$

Q. If $f\left(x\right)=x^3+4x^2+\mathrm{kx}+1$ is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then .

1. k = 4
2. k = 2
3. k = -1
4. k has no real value