$A$ tank with a rectangular base and rectangular sides,open at the top,is to be constructed such that its depth is $2 \ m$ and volume is $8 \ m^{3}$. If the cost of building the tank is Rs $70$ per square metre for the base and Rs $45$ per square metre for the sides,what is the minimum cost of the tank?

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

Let $f(x) = x^3 + px + 1$ and consider the following three statements:
$(i)$ For $p \geqslant 0$,$f(x) = 0$ has one negative root and $f(x)$ is monotonic.
$(ii)$ For $-1 < p < 0$,$f(x) = 0$ has one negative root and $f(x)$ is non-monotonic.
$(iii)$ For $p < -3/\sqrt[3]{4}$,$f(x) = 0$ has three real and distinct roots.
Which of the following is correct?

For all real $x$,the minimum value of the function $f(x)=\frac{1-x+x^2}{1+x+x^2}$ is

$A$ wire of length $2$ units is cut into two parts,which are bent respectively to form a square of side $x$ units and a circle of radius $r$ units. If the sum of the areas of the square and the circle so formed is minimum,then:

Given $\lambda \in [0, 20]$,find the number of integral values of $\lambda$ for which the function $f(x) = x^3 - 12x + \lambda$ has a point of local maxima.