In the interval $(0, 1)$,the maximum value of the function $f(x) = |x \ln x|$ is:

$A$ tank with a rectangular base and rectangular sides, open at the top is made. The depth of the tank is $4 \text{ m}$ and its volume is $36 \text{ m}^3$. The cost of the base material is $\text{Rs. } 100 \text{ per m}^2$ and the cost of the side material is $\text{Rs. } 50 \text{ per m}^2$. Find the minimum cost of the tank.

The minimum value of $\frac{\log x}{x}$ in the interval $(2, \infty)$ is

In an isosceles trapezium,the length of one of the parallel sides and the lengths of the non-parallel sides are all equal to $30$. In order to maximize the area of the trapezium,the smallest angle should be:

If $x = -1$ and $x = 2$ are the extreme points of the function $y = a \log|x| + bx^2 + x$,then find the values of $a$ and $b$.

Let $a > 0$. If the function $f(x) = 6x^3 - 45ax^2 + 108a^2x + 1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1x_2 = 54$,then $a + x_1 + x_2$ is equal to: