The number of real roots of the equation $e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^x+1=0$ is:

If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is _____.

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

Let $f:R\to R$ be defined as

$f\left(x\right)=\left\{\begin{array}{ll}-55x,& \mathrm{if} x<-5\\ 2x^3-3x^2-120x,& \mathrm{if}-5\le x\le 4\\ 2x^3-3x^2-36x-336,& \mathrm{if} x>4\end{array}\right.$

Let $A=\{x\in R:f$ is increasing}. Then $A$ is equal to: