Divide $10$ into two parts such that the sum of twice the first part and the square of the second part is minimum. The two parts are:

The function $f(x) = \int_1^x \{ 2(t - 1)(t - 2)^3 + 3(t - 1)^2(t - 2)^2 \} dt$ is maximum when $x$ is equal to

The sum of the absolute maximum and minimum values of the function $f(x) = |x^2 - 5x + 6| - 3x + 2$ in the interval $[-1, 3]$ is equal to:

$P$ and $Q$ are two points on a circle with center $C$ and radius $\alpha$. The angle $\angle PCQ = 2\theta$. The radius $r$ of the circle inscribed in the triangle $CPQ$ is maximum when:

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

Find the total number of local maxima and local minima of the function $f(x) = \begin{cases} (2+x)^3, & -3 < x \leq -1 \\ x^{2/3}, & -1 < x < 2 \end{cases}$