Let $$f(x) = \left\{ {\matrix{ { - 2\sin x,} & {if\,x \le - {\pi \over 2}} \cr {A\sin x + B,} & {if\, - {\pi \over 2} < x < {\pi \over 2}} \cr {\cos x} & {if\,x \ge {\pi \over 2}} \cr } } \right.$$. Then,

Consider the non-constant differentiable function f one one variable which obeys the relation $${{f(x)} \over {f(y)}} = f(x - y)$$. If f' (0) = p and f' (5) = q, then f' ($$-$$5) is

If f'' (0) = k, k $$ \ne $$ 0, then the value of

$$\mathop {\lim }\limits_{x \to 0} {{2f(x) - 3f(2x) + f(4x)} \over {{x^2}}}$$ is

Let $$f(x) = \left\{ {\matrix{ {{{{x^p}} \over {{{(\sin x)}^q}}},} & {if\,0 < x \le {\pi \over 2}} \cr {0,} & {if\,x = 0} \cr } } \right.$$, $$(p,q \in R)$$. Then, Lagrange's mean value theorem is applicable to f(x) in closed interval [0, x]