For $$a \in R,\,|a|\, > 1$$, let

$$\mathop {\lim }\limits_{n \to \infty } \left( {{{1 + \root 3 \of 2 + ...\root 3 \of n } \over {{n^{7/3}}\left( {{1 \over {{{(an + 1)}^2}}} + {1 \over {{{(an + 2)}^2}}} + ... + {1 \over {{{(an + n)}^2}}}} \right)}}} \right) = 54$$

Let f : R be a function. We say that f has

PROPERTY 1 if $$\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {\sqrt {|h|} }}$$ exists and is finite, and

PROPERTY 2 if $$\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {{h^2}}}$$ exists and is finite. Then which of the following options is/are correct?

Let f : R $$ \to $$ R be given by

$$f(x) = \left\{ {\matrix{ {{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 3x + 1,} & {x < 0;} \cr {{x^2} - x + 1,} & {0 \le x < 1;} \cr {{2 \over 3}{x^3} - 4{x^2} + 7x - {8 \over 3},} & {1 \le x < 3;} \cr {(x - 2){{\log }_e}(x - 2) - x + {{10} \over 3},} & {x \ge 3;} \cr } } \right\}$$

Then which of the following options is/are correct?

Let f : (0, $$\pi $$) $$ \to $$ R be a twice differentiable function such that $$\mathop {\lim }\limits_{t \to x} {{f(x)\sin t - f(t)\sin x} \over {t - x}} = {\sin ^2}x$$ for all x$$ \in $$ (0, $$\pi $$).

If $$f\left( {{\pi \over 6}} \right) = - {\pi \over {12}}$$, then which of the following statement(s) is (are) TRUE?