Monotonicity

Consider the function $f\left(x\right)=\frac{1}{x+2}$, then $f\left(x\right)$ is 

The function $f\left(x\right)=\log \left(1+x\right)-\frac{2x}{2+x}$ is increasing on

The interval in which $y=x^2{e}^{-x}$ is increasing is

The length of the longest interval, in which $f\left(x\right)=3\sin x-4{\sin }^{3}x$ is increasing, is

Let the function $f:R\to R$ be defined by $f\left(x\right)=2x+\sin x,x\in R$. Then $f$ is -

The function $g\left(x\right)=e^{x^2} \frac{\log (\pi +x)}{\log (e+x)} \left(x \ge 0\right)$ is:

In the interval $\left(1,\mathrm{ }2\right)$, function $f\left(x\right)=2\left|x-1\right|+3\left|x-2\right|$ is

The value of $k$ for which $f\left(x\right)=\sin x\mathrm{ }–\mathrm{ }\cos x – kx+b$ decreases for all real $x$ is given by:

If $f\left(x\right)=\frac{1}{x+1}-\log \left(1+x\right), x>0$, then $f$ is:

The value of $x$ for which the polynomial $2x^3-9x^2+12x+4$ is a decreasing function of $x$, is:

If $f\left(x\right)=\left(ab-b^2-2\right)x+{\int }_0^x({\cos }^4\theta +{\sin }^4\theta )d\theta$ is decreasing function of $x$ for all $x\in R$ and $b\in R, b$ being independent of $x$, then

If $f\left(x\right)=\sin x-\cos x$, the interval in which function is decreasing in $0\le x\le 2\pi$, is:

If $f\left(x\right)=x{\mathrm{e}}^{x(1-x)}$, then $f\left(x\right)$ is:

How many real solutions does the equation $x^7+14x^5+16x^3+30x-560=0$ have?

The function $f\left(x\right)={\cot }^{-1}x+x$ is increasing in the interval:

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

If $f\left(x\right)={\int }_{x^2}^{x^2+1}{e}^{-t^2}dt$, then $f\left(x\right)$ increases in

For a given integer $k$, in the interval $\left[2\pi k-\frac{\pi }{2}, 2\pi k+\frac{\pi }{2}\right]$ the graph of $\sin x$ is

The function $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$ is increasing when

Define $f\left(x\right)={\int }_0^x\sin t dt, x\ge 0$. Then,