For all real $x$,the minimum value of the function $f(x)=\frac{1-x+x^2}{1+x+x^2}$ is

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:

$f(x) = \int\limits_0^x t(t - 1)(t - 2) dt$ takes on its minimum value when:

If the set of all values of $a$,for which the equation $5x^3 - 15x - a = 0$ has three distinct real roots,is the interval $(\alpha, \beta)$,then $\beta - 2\alpha$ is equal to . . . . . . .

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

The radius (in $cm$) of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone with radius $10 \ cm$ is