The minimum value of the function $f(x) = x \log x$ is

The triangle of maximum area that can be inscribed in a given circle of radius $r$ is ...... .

If $m$ and $M$ are respectively the absolute minimum and absolute maximum values of a function $f(x)=2x^3+9x^2+12x+1$ defined on $[-3,0]$, then $m+M=$

Let $a > 0$. If the function $f(x) = 6x^3 - 45ax^2 + 108a^2x + 1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1x_2 = 54$,then $a + x_1 + x_2$ is equal to:

The number of turning points of the curve $f(x) = 2 \cos x - \sin 2x$ in the interval $[-\pi, \pi]$ is

Let $f(x) = 2 \cos^{-1} x + 4 \cot^{-1} x - 3x^2 - 2x + 10$,where $x \in [-1, 1]$. If $[a, b]$ is the range of the function,then $4a - b$ is equal to: