STATEMENT - 1: $$\mathop {\lim }\limits_{x \to 0} \,\,\left[ {g\left( x \right)\cot x - g\left( 0 \right)\cos ec\,x} \right] = f''\left( 0 \right)$$ and

STATEMENT - 2: $$f'\left( 0 \right) = g\left( 0 \right)$$

Let a and b be non-zero real numbers. Then, the equation

$$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$$ represents :

Let $$g(x) = {{{{(x - 1)}^n}} \over {\log {{\cos }^m}(x - 1)}};0 < x < 2,m$$ and $$n$$ are integers, $$m \ne 0,n > 0$$, and let $$p$$ be the left hand derivative of $$|x - 1|$$ at $$x = 1$$. If $$\mathop {\lim }\limits_{x \to {1^ + }} g(x) = p$$, then

The total number of local maxima and local minima of the function

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