If $a_n$ is the greatest term in the sequence $a_n = \frac{n^3}{n^4+147}$,$n = 1, 2, 3, \ldots$,then $n$ is equal to $..........$.

The function $f(x) = x^{-x}, (x \in R)$ attains a maximum value at $x =$

If the distance $s$ covered in time $t$ by a particle moving on a straight line is given by $s = t^5 - 40t^3 + 30t^2 + 80t - 250$,then its minimum acceleration is

Find the local maximum and local minimum values for the function $f(x) = x\sqrt{1 - x}$ where $0 < x < 1$.

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 local maximum value $l$ and local minimum value $m$ of $f(x) = \frac{x^2+2x+2}{x+1}$ in $R - \{-1\}$ exist at $\alpha, \beta$ respectively, then $\frac{l+m}{\alpha+\beta} =$