The sufficient conditions for the function $f:R \to R$ to have a local maximum at $x = a$ are:

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:

The range of $a \in R$ for which the function $f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)$ where $x \neq 2 n \pi, n \in N$,has critical points,is

The number $10$ is divided into two parts such that the sum of double of the first part and the square of the second part is minimum. The two parts are respectively:

Let $f(x) = \begin{cases} 3 - x, & 0 \le x < 1 \\ x^2 + \log_e b, & x \ge 1 \end{cases}$. The set of values of $b$ such that $f(x)$ has a local minimum at $x = 1$ is

Maximum value of the function $f(x) = [x(x-1) + 1]^{\frac{1}{3}}$ for $0 \leq x \leq 1$ is . . . . . . .