If $x - 2y = 4,$ the minimum value of $xy$ is

In the interval $(-4, 4)$,the function $f(x) = \int_{-10}^x (t^4 - 4)e^{-4t} dt$ has:

If $x$ is real,then the minimum value of $\frac{x^2-x+1}{x^2+x+1}$ is

If the function $f(x)=2x^{3}-9ax^{2}+12a^{2}x+1$ where $a>0$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2}=q$,then $a$ is equal to

If the curved surface area of a right circular cylinder inscribed in a sphere of radius $22 \ cm$ is maximum,then the height of the cylinder will be:

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then: