The two positive numbers with sum $t$,and the sum of their squares is minimum are

The condition for the function $f(x) = x^3 + px^2 + qx + r$ $(x \in R)$ to have no extreme value is:

The function $f$ is defined by $f(x) = x^p (1 - x)^q$ for all $x \in R$,where $p, q$ are positive integers. The function has a maximum value for $x$ equal to:

If $A=\{x : 9x \geq x^2+20\}$ and $f: A \rightarrow R$ is defined by $f(x)=2x^3-15x^2+36x-48$,then the maximum value of $f(x)$ 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 $m$)

Let $S=(-1, \infty)$ and $f: S \rightarrow R$ be defined as $f(x)=\int_{-1}^x (e^t-1)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61} dt$. Let $p$ be the sum of the squares of the values of $x$ where $f(x)$ attains local maxima on $S$,and $q$ be the sum of the values of $x$ where $f(x)$ attains local minima on $S$. Then,the value of $p^2+2q$ is