If $y = \alpha \log x + \beta x^3 - x$ has extreme values at $x = -1$ and $x = 1$,then $\alpha$ and $\beta$ are respectively

The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is . . . . . . .

$20 \ m$ of wire is available to fence a flowerbed in the form of a circular sector. If the flowerbed is to have maximum surface area,then the radius of the circle is (in $m$)

If $x + 2y = 8$,then the maximum value of $xy$ is .......

Let $f(x) = x^3 + ax^2 + bx + c$ where $a, b, c$ are real numbers. If $f(x)$ has a local minimum at $x = 1$ and a local maximum at $x = -\frac{1}{3}$ and $f(2) = 0$,then $\int_{-1}^1 f(x) dx$ equals

Let $AD$ and $BC$ be two vertical poles at $A$ and $B$ respectively on a horizontal ground. If $AD = 8 \ m$,$BC = 11 \ m$ and $AB = 10 \ m$; then the distance (in meters) of a point $M$ on $AB$ from the point $A$ such that $MD^{2} + MC^{2}$ is minimum is