If $f(x)=x^5-5 x^4+5 x^3-10$ has its local maxima and minima at $x=a$ and $x=b$ respectively,then $2 a+b$ is equal to

An open tank with a square bottom,to contain $4000 \ cm^3$ of liquid,is to be constructed. The dimensions of the tank,so that the surface area of the tank is minimum,are

If $x=-2$ and $x=4$ are the extreme points of $y=x^3-\alpha x^2-\beta x+5$,then

Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x-2) + \alpha x^2 + 4x - \alpha, x > 2$, is

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

In a regular triangular prism,the distance from the centre of one base to one of the vertices of the other base is $l$. Find the altitude of the prism for which the volume is greatest.