If the petrol burnt in driving a motor boat varies as the cube of the velocity,then the speed (in $km/h$) of the boat relative to the water,going against a water flow of $C \ km/h$,so that the quantity of petrol burnt is minimum,is:

For the function $f(x) = x \cos \frac{1}{x}, \quad x \geq 1$,consider the following statements:
$(A)$ For at least one $x$ in the interval $[1, \infty), f(x+2)-f(x) < 2$
$(B)$ $\lim _{x \rightarrow \infty} f^{\prime}(x) = 1$
$(C)$ For all $x$ in the interval $[1, \infty), f(x+2)-f(x) > 2$
$(D)$ $f^{\prime}(x)$ is strictly decreasing in the interval $[1, \infty)$
Which of the following combinations of statements is correct?

If $20$ is divided into two parts such that the product of the cube of one part and the square of the other part is maximum,then these two parts are:

Prove that the function $g(x) = \log x$ does not have any maxima or minima.

The maximum volume of a right circular cylinder if the sum of its radius and height is $6 \text{ m}$ is: (in $\pi \text{ m}^3$)

Let $f: R \rightarrow R$ be given by $f(x) = |x^{2} - 1|$, $x \in R$. Then: