If $f(x) = \frac{\sin(x + a)}{\sin(x + b)}$,$a \neq b$,then $f$ is......

If $y = a \log x + b x^2 + x$ has its extreme values at $x = -1$ and $x = 2$,then the value of $\left(\frac{a}{b} + \frac{b}{a}\right)$ is

$A$ box is to be made with a square base and an open top. If the area of the material used is $48 \, m^2$,then the maximum volume of the box is ........... $m^3$.

$A$ closed vessel tapers to a point both at its top $E$ and its bottom $F$ and is fixed with $EF$ vertical. When the depth of the liquid in it is $x \, \text{cm}$,the volume of the liquid in it is $V(x) = x^2 (15 - x) \, \text{cu. cm}$. The length $EF$ is ........ $\text{cm}$.

$A$ wire of length $20 \ m$ is to be cut into two pieces. $A$ piece of length $\ell_1$ is bent to make a square of area $A_1$ and the other piece of length $\ell_2$ is made into a circle of area $A_2$. If $2A_1 + 3A_2$ is minimum,then $(\pi \ell_1) : \ell_2$ is equal to:

The condition for the function $f(x) = x^3 + px^2 + qx + r$ $(x \in R)$ to have no extreme value is: