The maximum value of $xy$ subject to $x + y = 8$ is

If $f(x) = x^3 + ax^2 + bx + c$ has a minimum at $x = 3$ and a maximum at $x = -1$,then:

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$:

The figure shown below is the graph of the derivative of some function $y=f(x)$. Then,

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

$A$ stone is thrown vertically upwards from the top of a tower $64 \ m$ high according to the law $s=48t-16t^{2}$. The greatest height attained by the stone above the ground is (in $m$)