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 $....$

The minimum value of the function $f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ on the interval $[0, 3]$ is equal to:

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x$,then

Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4$ for all $x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln(g(x))$ for all $x \in R$,then the number of points in $R$ at which $g$ has a local maximum is

Let $f$ be a differentiable function satisfying $f(x)=1-2x+\int_{0}^{x}e^{(x-t)}f(t)dt, x\in R$ and let $g(x)=\int_{0}^{x}(f(t)+2)^{15}(t-4)^{6}(t+12)^{17}dt, x\in R.$ If $p$ and $q$ are respectively the points of local minima and local maxima of $g$, then the value of $|p+q|$ is equal to . . . . . . .

Let $A$ be the region enclosed by the parabola $y^2=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $A$ is ..................