Condition for Monotonically Decreasing Function

Q. The function $f\left(x\right)=\frac{x+1}{x^3+1}$ can be written as the sum of an even function $g\left(x\right)$ and an odd function $h\left(x\right)$. Then the value of $|g\left(0\right)|$ is

Q. Consider a real-valued continuous function $f\left(x\right)$ defined on the interval $[a,b].$ Which of the following statements does NOT hold good?

1. If $f\left(x\right)\ge 0$ on $[a,b],$ then $\underset{a}{\overset{b}{\int }}f\left(x\right)dx\le \underset{a}{\overset{b}{\int }}{\left(f\right(x\left)\right)}^{2}dx$
2. If $f\left(x\right)$ is increasing on $[a,b],$ then $\left(f\right(x){)}^2$ is increasing on $[a,b]$
3. If $f\left(x\right)$ is increasing on $[a,b],$ then $f\left(x\right)\ge 0$ on $(a,b)$
4. If $f\left(x\right)$ attains a minimum at $x=c$ where $a<c<b,$ then $f^{\prime }\left(c\right)=0$

Q.

If f(x) , g(x) and h(x) are three differentiable functions throughout their domains and given that their first derivatives are -

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

$g^{\prime }\left(x\right)=0$

$h^{\prime }\left(x\right)\le 0$

throughout their domains. Then choose the correct option of monotonically decreasing functions -

1. f(x) & h(x)

2. g(x) & h(x)

3. f(x), g(x) & h(x)

4. Only h(x)

Q. Let $h$ be a twice differentiable positive function on an open interval $J.$ Let $g\left(x\right)=\ln \left(h\right(x\left)\right)$ for each $x\in J.$ Suppose ${\left(h^{\prime }\right(x\left)\right)}^2>h^{\prime \prime }\left(x\right)h\left(x\right)$ for each $x\in J.$ Then

1. $g$ is increasing on $J$
2. $g$ is decreasing on $J$
3. $g$ is concave up on $J$
4. $g$ is concave down on $J$

Q. Find the second order derivative of the function $y=\log x$

Q.

Which of the following is a monotonically decreasing function?

1. f(x) = ln (x)

2. f(x) = 3 - x³

3. f(x) = e^x

4. f(x) = 1/ x²

Q. The functions $f\left(x\right)$ and $g\left(x\right)$ are positive and continuous. If $f\left(x\right)$ is increasing and $g\left(x\right)$ is decreasing, then $\underset{0}{\overset{1}{\int }}f\left(x\right)\left[g\right(x)-g(1-x\left)\right]dx$

1. is always non-positive
2. is always non-negative
3. can take positive or negative values
4. can't be determined

Q. Two functions $f\left(x\right)$ and $g\left(x\right)$ are defined as $f\left(x\right)=\{\begin{array}{cc}[x{]}^{\left\{x\right\}},& x\ne 0\\ 0,& x=0\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}\{x{\}}^{\left[x\right]},& x\ne 0\\ 0,& x=0\end{array}$ (where $[.]$ and $\{.\}$ refer to the greatest integer function and the fractional part function respectively and $n\in N).$ Then which of the following statements is/are TRUE?

1. $\underset{0}{\overset{n+1}{\int }}f\left(x\right)dx-\underset{0}{\overset{n}{\int }}f\left(x\right)dx=\frac{n-1}{\ln n}$
2. $\underset{0}{\overset{n+1}{\int }}g\left(x\right)dx-\underset{0}{\overset{n}{\int }}g\left(x\right)dx=\frac{1}{n+1}$
3. $\underset{0}{\overset{t}{\int }}f\left(x\right)dx=\underset{0}{\overset{t}{\int }}g\left(x\right)dx$ has at least one solution in $t\in (0,\frac{\pi }{2})$
4. $e<\underset{n\to \infty }{lim}\underset{0}{\overset{n}{\int }}g\left(x\right)dx<e^2$

Q. Let $f$ be a continuous function. If $\underset{0}{\overset{x}{\int }}f\left(t\right)dt=e^x-c{e}^{2x}\underset{0}{\overset{1}{\int }}f\left(t\right)e^{-t}dt,$ where $c$ is a non-zero constant, then

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

Q.

Which of the following is a monotonically decreasing function?

1. f(x) = ln (x)

2. $f\left(x\right)=3-x^3$

3. $f\left(x\right)=e^x$

4. $f\left(x\right)=\frac{1}{x^2}$

Q. If $f\left(x\right)$ and $g\left(x\right)$ are continuous functions, then the value of $\underset{\ln \lambda }{\overset{\ln 1/\lambda }{\int }}\frac{f\left(\frac{x^2}{4}\right)\left[f\right(x)-f(-x\left)\right]}{g\left(\frac{x^2}{4}\right)\left[g\right(x)+g(-x\left)\right]}dx$ is

1. dependent on $\lambda$
2. a non zero constant
3. zero
4. ${\lambda }^2$

Q. $f\left(x\right)=\frac{x}{lo{g}_{e}x},x\ne 1$, is decreasing in interval

1. None of these
2. (0, e)
3. (1, e)
4. (e, ∞)

Q. If the function $e^{-x}f\left(x\right)$ assumes its minimum in the interval $[0,1]\text{at}x=\frac{1}{4},$ which of the following is true?

1. $f^{\prime }\left(x\right)<f\left(x\right),0<x<\frac{1}{4}$
2. $f^{\prime }\left(x\right)<f\left(x\right),\frac{1}{4}<x<\frac{3}{4}$
3. $f^{\prime }\left(x\right)<f\left(x\right),\frac{3}{4}<x<1$
4. $f^{\prime }\left(x\right)>f\left(x\right),0<x<\frac{1}{4}$

Q. $f\left(x\right)=\frac{x}{lo{g}_{e}x},x\ne 1$, is decreasing in interval

1. None of these
2. (1, e)
3. (e, ∞)
4. (0, e)

Q. If $f:R\to R$ is a continuous function and satisfies $f\left(x\right)=e^x+\underset{0}{\overset{1}{\int }}{e}^{x}f\left(t\right)dt$, then

1. $f\left(0\right)<0$
2. $f\left(x\right)$ is a decreasing function $\forall x\in R$
3. $f\left(x\right)$ is a increasing function $\forall x\in R$
4. $\underset{0}{\overset{1}{\int }}f\left(x\right)dx>0$

Q. If $p=\frac{1}{{\log }_3\pi }+\frac{1}{{\log }_4\pi }+1$ then

1. $2.5<p<3$
2. $p>3$
3. $1.5<p<2$
4. $2<p<2.5$

Q. Function $f\left(x\right)=x^3-3x^2-9x+22$ is monotonic decreasing for

1. $-1<x<3$
2. $1<x<3$
3. $-3<x<1$
4. None of these

Q. Let $f:R\to R$ be a function which satisfies $f(x+y)=f\left(x\right)+f\left(y\right)\forall x,y\in R$. If $f\left(1\right)=2$ and $g\left(n\right)=\sum _{k=1}^{(n-1)}f\left(k\right),n\in N$ then the value of $n$, for which $g\left(n\right)=20$, is:

1. $9$
2. $5$
3. $4$
4. $20$

Q.

If f(x) , g(x) and h(x) are three differentiable functions throughout their domains and given that their first derivatives are -

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

$g^{\prime }\left(x\right)=0$

$h^{\prime }\left(x\right)\le 0$

throughout their domains. Then choose the correct option of monotonically decreasing functions -

1. Only h(x)

2. f(x) & h(x)

3. g(x) & h(x)

4. f(x), g(x) & h(x)

Q. Let $h$ be a twice differentiable positive function on an open interval $J.$ Let $g\left(x\right)=\ln \left(h\right(x\left)\right)$ for each $x\in J.$ Suppose ${\left(h^{\prime }\right(x\left)\right)}^2>h^{\prime \prime }\left(x\right)h\left(x\right)$ for each $x\in J.$ Then

1. $g$ is increasing on $J$
2. $g$ is decreasing on $J$
3. $g$ is concave up on $J$
4. $g$ is concave down on $J$

Q. The domain and range of the real function f defined by $f\left(x\right)=\frac{4-x}{x-4}$ is given by

1. Domain$=R-\{-4\}$, Range $=\{-1,1\}$
2. Domain $=R-\left\{1\right\},$ Range $R$
3. Domain $=R,$ Range $=\{–1,1\}$
4. Domain$=R-\left\{4\right\}$, Range $=\{-1\}$

Q. $f\left(x\right)=\frac{x}{lo{g}_{e}x},x\ne 1$, is decreasing in interval

1. $(0,e)$
2. $(1,e)$
3. $(e,\infty )$
4. R

Q. Check monotonicity of f(x) at indicated point:
1) $f\left(x\right)=x^{1/3}$ at $x=0$

Q.

Which of the following is a monotonically decreasing function?

1. f(x) = ln (x)

2. $f\left(x\right)=\frac{1}{x^2}$

3. $f\left(x\right)=3-x^3$

4. $f\left(x\right)=e^x$

Q. Let $f$ be a real valued function, defined on $R-\{-1,1\}$ and given by $f\left(x\right)=3{\log }_e\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}.$ Then in which of the following intervals, function $f\left(x\right)$ is increasing ?

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

Q. If $f:R\to R$ is a continuous function and satisfies $f\left(x\right)=e^x+\underset{0}{\overset{1}{\int }}{e}^{x}f\left(t\right)dt$, then

1. $f\left(0\right)<0$
2. $f\left(x\right)$ is a decreasing function $\forall x\in R$
3. $f\left(x\right)$ is a increasing function $\forall x\in R$
4. $\underset{0}{\overset{1}{\int }}f\left(x\right)dx>0$

Q. $S_1:$ If $f\left(x\right)$ is an increasing function with downward concavity, then concavity of $f^{-1}\left(x\right)$ is upwards.
$S_2:$ If $f\left(x\right)$ is decreasing function with downward concavity, then concavity of $f^{-1}\left(x\right)$ is upwards.

1. $S_1$ and $S_2$ both are true
2. $S_1$ is true and $S_2$ is false
3. $S_1$ is false and $S_2$ is true
4. $S_1$ and $S_2$ both are false

Q. Let $f:[\frac{1}{2},1]\to 1$ (the set of all real numbers) be a positive, non-constant and differentiable function such that f(x) < 2f(x) and $f\left(\frac{1}{2}\right)=1$. Then the value of ${\int }_{1/2}^{1}f\left(x\right)dx$ lies in the interval

1. $(0,\frac{e-1}{2})$
2. $(\frac{e-1}{2},e-1)$
3. $(e-1,2e-1)$
4. $(2e-1,2e)$

Q. $f\left(x\right)=x^3-27x+5$ is monotonically increasing for

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

Q. Let $f$ be a real valued function, defined on $R-\{-1,1\}$ and given by $f\left(x\right)=3{\log }_e\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}.$ Then in which of the following intervals, function $f\left(x\right)$ is increasing ?

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