Condition for Strictly Increasing

Q.

The maximum value of $f\left(x\right)=\frac{x}{\left[4+x+x^2\right]}$ on $\left[-1,1\right]$ is

1. $-\left(\frac{1}{3}\right)$

2. $-\left(\frac{1}{4}\right)$

3. $\frac{1}{4}$

4. $\frac{1}{6}$

Q. Let $f\left(x\right)=e^x-x$ and $g\left(x\right)=x^2-x,\forall x\in R$. Then the set of all $x\in R$, where the function $h\left(x\right)=(f\circ g)\left(x\right)$ is increasing, is :

1. $[-\frac{1}{2},0]\cup [1,\infty )$
2. $[-1,-\frac{1}{2}]\cup [\frac{1}{2},\infty )$
3. $[0,\infty )$
4. $[0,\frac{1}{2}]\cup [1,\infty )$

Q. Let f be a differentiable function on
R and satisfying the integral equation ${\int }_0^{x}f\left(t\right)dt+{\int }_0^{x}t.f(x-t)dt=-1+e^{-x}\text{for all}xϵR,$ then

1. $f^{\prime }\left(0\right)=2$
2. $f\left(2\right)=e^{-2}$
3. $f\left(0\right)+f^{\prime }\left(0\right)=1$
4. $f^{\prime }\left(0\right)=1$

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

• 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 INCORRECT combination?

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

Q.

Let $f\left(x\right)=x\left|x\right|$ The set of points where $f\left(x\right)$ is twice differentiable be:

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

2. $(1,\infty )$

3. $Z–\left\{0\right\}$

4. $R-\left\{0\right\}$

Q. Find $\frac{dy}{dx}$ if $x^y+y^x=1$

Q. Let $f:[a,b]\to R$ be such that $f$ is differentiable in $(a,b),f$ is continuous at $x=a$ and $x=b$ and moreover $f\left(a\right)=0=f\left(b\right)$. Then

1. there exists atleast one point $c$ in $(a,b)$ such that $f^{\prime }\left(c\right)=f\left(c\right)$
2. $f^{\prime }\left(x\right)=f\left(x\right)$ does not hold at any point in $(a,b)$
3. at every point of $(a,b),f^{\prime }\left(x\right)>f\left(x\right)$
4. at every point of $(a,b),f^{\prime }\left(x\right)<f\left(x\right)$

Q. If $xdy=y(dx+ydy),y\left(1\right)=1$ and$y\left(1\right)=1$, $y\left(x\right)>0$. Then, $y(-3)$ is equal to

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

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

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

Q. If $f:R\to R$ is a differentiable function such that $f^{\prime }\left(x\right)>2f\left(x\right)$ for all $x\in R$, and $f\left(0\right)=1,$ then

1. $f\left(x\right)$ is increasing in$(0,\infty )$
2. $f\left(x\right)$ is decreasing in$(0,\infty )$
3. $f\left(x\right)>e^{2x}$ in $(0,\infty )$
4. $f^{\prime }\left(x\right)<e^{2x}$ in $(0,\infty )$

Q. If $y\frac{dy}{dx}=x[\frac{y^2}{x^2}+\frac{\varphi \left(\frac{y^2}{x^2}\right)}{{\varphi }^{\prime }\left(\frac{y^2}{x^2}\right)}],$ $x>0,$ $\varphi >0$ and $y\left(1\right)=-1,$ then $\varphi \left(\frac{y^2}{4}\right)$ is equal to

1. $4\varphi \left(2\right)$
2. $4\varphi \left(1\right)$
3. $2\varphi \left(1\right)$
4. $\varphi \left(1\right)$

Q. The minimum value of $f\left(x\right)=x+\frac{4}{x+2},x\ge 0$ is

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

Q. If $f\left(x\right)$ is a function such that $f^{\prime }\left(x\right)=(x-1{)}^2(4-x)$, then

1. $f\left(0\right)=0$
2. $f\left(x\right)$ is increasing in $(0,3)$
3. $x=4$ is a critical point of $f\left(x\right)$
4. $f\left(x\right)$ is decreasing in (3, 5)

Q. If $f$ is a real-valued differentiable function satisfying $|f\left(x\right)-f\left(y\right)|\le {(x-y)}^2$, $x,yϵR$ and $f\left(0\right)=0$, then $f\left(1\right)$ equals

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

Q. The number of real solution of $x-\frac{1}{x^2-4}=2-\frac{1}{x^2-4}$, is:

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

Q.

What is the condition for a strictly increasing differentiable function?

1. $f^{\prime }\left(x\right)>0$

2. $f^{\prime }\left(x\right)\ge 0$

3. $f^{\prime }\left(x\right)<0$

4. $f^{\prime }\left(x\right)\le 0$

Q. The function $y=f\left(x\right)$, defined parametrically as $x=2t-|t-1|$ and $y=2t^2+t\left|t\right|$, is

1. Continuous and differentiable for $x\in R$
2. None of these
3. Continuous for $x\in R$ and differentiable for $x\in R-\text{{-1, 2}}$
4. Continuous for $x\in R$ and differentiable for $x\in R-\text{{2}}$

Q. If $\varphi$ is a differentiable function with $\varphi \left(0\right)=0$ and ${\varphi }^{\prime }\left(x\right)+2\varphi \left(x\right)\le 1$ then maximum value of $\varphi \left(x\right)$ is

Q. Let $f\left(x\right)=e^x-e^{-x},g\left(x\right)=\ln (x^2-5x+6),$ then which of the following(s) is(are) correct about $h\left(x\right)=f\left(g\right(x\left)\right)$

1. increasing in $(2,\infty )$
2. increasing in $(3,\infty )$
3. decreasing in $(-\infty ,2)$
4. decreasing in $(-\infty ,\frac{5}{2})$

Q. Let f be a given function such that f(x)>0, ∀x \in D_f. Let \)
$I_1{\int }_{1-k}^{k}xf\left(x\right(1-x\left)\right)dx$
$I_2={\int }_{1-k}^{k}f\left(x\right(1-x\left)\right)dx$, where $2k-1>0$
then $\frac{I_1}{I_2}$ is equal to

1. $1$
2. $k$
3. $\frac{1}{2}$
4. $2$

Q. Let $f:[0,2]\to R$ be a twice differentiable function such that $f^{\prime \prime }\left(x\right)>0$, for all $x\in (0,2)$. If $\varphi \left(x\right)=f\left(x\right)+f(2-x),$ then $\varphi$ is:

1. decreasing on $(0,2)$
2. decreasing on $(0,1)$ and increasing on $(1,2)$
3. increasing on $(0,2)$
4. increasing on $(0,1)$ and decreasing on $(1,2)$

Q. The interval in which $f\left(x\right)={\int }_0^x\left\{\right(t+1\left)\right(e^t-1\left)\right(t-2\left)\right(t+4\left)\right\}dt$ increases and decreases.

1. Increases on $(-\infty ,-4)\cup (-1,0)\cup (2,\infty )$ and decreases on $(-4,1)\cup (0,2)$
2. Increases on $(-\infty ,-4)\cup (-1,2)$ and decreases on $(-4,1)\cup (2,\infty )$
3. Increases on $(-\infty ,-4)\cup (2,\infty )$ and decreases on $(-4,2)$
4. Increases on $(-4,-1)\cup (0,2)$ and decreases on $(\infty ,-4)\cup (-1,0)\cup (2,\infty )$

Q. If Y is the solution of $(1+t)\frac{dy}{dt}-ty=1$ and Y(0) = -1 then y(1) is equal to

1. $e+\frac{-1}{2}$
2. $\frac{-1}{2}$
3. $e-\frac{-1}{2}$
4. $\frac{1}{2}$

Q. Let $f\left(x\right)=(x-1{)}^4\cdot (x-2{)}^n,nϵN.$ Then $f\left(x\right)$ has

1. a maximum at $x=1$ if $n$ is even
2. a maximum at $x=1$ if $n$ is odd
3. a minimum at $x=2$ if $n$ is even
4. a maximum at $x=2$ if $n$ is odd

Q.

What is the condition for a strictly increasing differentiable function?

1. f’(x) > 0

2. f’(x) ≥ 0

3. f’(x) < 0

4. f’(x) ≤ 0

Q. Find all the points of discontinuity of $f$ defined by $f\left(x\right)=|x|-|x+1|$.

Q. Let $g\left(x\right)=\frac{1}{4}f(2x^2-1)+\frac{1}{2}f(1-x^2)$ for all $x\in R,$ where $f^{\prime \prime }\left(x\right)>0\forall x\in R.$ If $g\left(x\right)$ is necessarily strictly increasing in the interval $(a,0)\cup (b,\infty ),$ then the value of $(a+b)$ is

Q. $$\begin{array}{cc}\text{List I}& \text{List II}\\ \text{(A) If f satisfies}|f\left(u\right)-f\left(v\right)|\le |u-v|\text{for}u\&v\text{in}[a,b]& \\ \text{, then}\text{maximum possible value of}\left|\underset{2}{\overset{4}{\int }}f\left(x\right)dx-f\left(2\right)dx\right|\text{is}& \text{(P)}1\\ \text{(B)}\text{Let}f\left(z\right)\text{being a complex function defined}& \\ \text{as}f\left(z\right)=\frac{az+b}{cz+d}\text{, where}a,b,c,d& \\ \text{are non-zero real numbers. If}f\left(z_1\right)=f\left(z_2\right)& \\ \text{for all}{z}_1\ne z_2\text{and}b,a,c& \\ \text{are in G.P., then the value of}\frac{a}{d}\text{is}& \text{(Q)}2\\ \text{(C) Evaluate:}\underset{x\to 5}{lim}\frac{1-\cos (x^2-9x+20)}{(x-5{)}^2}& \text{(R)}\frac{1}{2}\\ \text{(D) If}\text{The number of values of (a) that satisfying}& \\ \underset{x\to -a}{lim}\frac{x^5+a^5}{x+a}=5\text{is}& \text{(S)}4\\ & \text{(T)}5\\ & \text{(U)}3\end{array}$$

Which of the following is CORRECT option ?

1. $\left(B\right)\to \left(S\right)$
2. $\left(C\right)\to \left(T\right)$
3. $\left(D\right)\to \left(R\right)$
4. $\left(A\right)\to \left(Q\right)$

Q. Which of the following is true for $0<x<1$?

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

Q. Let $f:[0,2]\to R$ be a twice differentiable function such that $f^{\prime \prime }\left(x\right)>0$, for all $x\in (0,2)$. If $\varphi \left(x\right)=f\left(x\right)+f(2-x),$ then $\varphi$ is:

1. increasing on $(0,1)$ and decreasing on $(1,2)$
2. decreasing on $(0,1)$ and increasing on $(1,2)$
3. increasing on $(0,2)$
4. decreasing on $(0,2)$