Let $$f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & {0 \le x < 1} \cr {2x - 3} & {1 \le x \le 3} \cr } } \right.$$

Find all possible real values of $$b$$ such that $$f(x)$$ has the smallest value at $$x=1$$.

In this questions there are entries in columns $$I$$ and $$II$$. Each entry in column $$I$$ is related to exactly one entry in column $$II$$. Write the correct letter from column $$II$$ against the entry number in column $$I$$ in your answer book.

Let the functions defined in column $$I$$ have domain $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$

$$\,\,\,\,$$Column $$I$$
(A) $$x + \sin x$$
(B) $$\sec x$$

$$\,\,\,\,$$Column $$II$$
(p) increasing
(q) decreasing
(r) neither increasing nor decreasing

A cubic $$f(x)$$ vanishes at $$x=2$$ and has relative minimum / maximum at $$x=-1$$ and $$x = {1 \over 3}$$ if $$\int\limits_{ - 1}^1 {f\,\,dx = {{14} \over 3}} $$, find the cubic $$f(x)$$.

What normal to the curve $$y = {x^2}$$ forms the shortest chord?