If $y = \frac{ax - b}{(x - 1)(x - 4)}$ has a turning point $P(2, -1)$,then the values of $a$ and $b$ are

Let $f(x)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\mathop {\lim }\limits_{x \to 0} \left(2+\frac{f(x)}{x^{3}}\right)=4,$ then which one of the following is not true?

If a running track of $500 \ ft$. is to be laid out enclosing a playground,the shape of which is a rectangle with a semicircle at each end,then the length of the rectangular portion such that the area of the rectangular portion is to be maximum is (in feet).

Statement-$I$: The sequence $a_n = \frac{n^2}{n^3 + 200}, n \in N$ has its $7^{th}$ term as the largest term.
Statement-$II$: The function $f(x) = \frac{x^2}{x^3 + 200}$ attains a local maximum at $x = 7$.

The minimum value of $f(x) = x + \frac{4}{x + 2}$ for $x > -2$ is

If the absolute maximum and absolute minimum values of the function $f(x) = x^3 - 2x^2 + x - 3$ defined on $[0, 2]$ are $M$ and $m$ respectively, then $M + m =$