The following figure shows the graph of a differentiable function $y=f(x)$ on the interval $[a, b]$ (not containing $0$). Let $g(x)=\frac{f(x)}{x}$. Which of the following is a possible graph of $y=g(x)$?

The perimeter of a sector is $p$. The area of the sector is maximum when its radius is

Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4$ for all $x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln(g(x))$ for all $x \in R$,then the number of points in $R$ at which $g$ has a local maximum is

The real number which most exceeds its cube is

$A$ 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?

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$: