The difference between the greatest and the least values of the function $f(x) = x(\ln x - 2)$ on the interval $[1, e^2]$ is:

If $x+y=6, x \geqslant 0, y \geqslant 0$,then the maximum value of $x^2 y$ is

Let a function $f:[0,5] \rightarrow R$ be continuous. $f(1)=3$ and $F$ be defined as $F(x)=\int_{1}^{x} t^{2} g(t) dt$,where $g(t)=\int_{1}^{t} f(u) du$. Then for the function $F$,the point $x=1$ is

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given,show that the area of the triangle is maximum when the angle between them is $\frac{\pi}{3}$.

Find the local maximum and local minimum values of the function $f$ given by $f(x) = 3x^4 + 4x^3 - 12x^2 + 12$.

If $0 < x < \frac{\pi}{2}$,then the maximum area (in sq. units) of the triangle whose vertices are $(0,0)$,$(x, \cos x)$ and $(\sin^3 x, 0)$ is