The two parts of $100$ for which the sum of double of the first part and the square of the second part is minimum,are

An open tank with a square bottom is to contain $4000 \ cm^3$ of liquid. Find the dimensions of the tank such that the surface area of the tank is minimum.

$A$ manufacturer sells $x$ items at a price of rupees $\left(6-\frac{x}{40}\right)$ each. The cost price of $x$ items is Rs. $\left(\frac{x}{5}+193\right)$. The maximum profit in Rs. is

The number of values of $x$ where the function $f(x) = \cos x + \cos (\sqrt{2} x)$ attains its maximum is

The displacement $x$ of a particle at time $t$ is given by $x = t^4 - kt^3$. If the velocity of the particle is maximum at $t = 2$,then $k = $ ..........

Let a function $f(x) = \begin{cases} -\ln(3x - [3x]) & ; 3x \neq n, n \in N \\ \ln(\operatorname{sgn}(3x)) & ; 3x = n, n \in N \end{cases}$,where $[.]$ and $\operatorname{sgn}(x)$ denote the greatest integer function and signum function respectively. Then the number of points where $f(x)$ is minimum in $x \in (0, 5)$ is: