If one of the vertices of the square circumscribing the circle $$|z-1|=\sqrt{2}$$ is $$(2+\sqrt{3 i})$$. Find the other vertices of square.

If $$f(x-y)=f(x) \circ g(y)-f(y) \circ g(x)$$ And $$g(x-y) =g(x) \circ g(y)+f(x) \circ f(y)$$ for all $$x, y \in \mathrm{R}$$. If right-hand derivative at $$x=0$$ exists for $$f(x)$$, find the derivative of $$g(x)$$ at $$x=0$$

If $$p(x)$$ be a polynomial of degree 3 satisfying $$p(-1)=10, p(1)=-6$$ and $$p(x)$$ has maximum at $$x=-1$$ and $$p'(x)$$ has minima at $$x=1$$. Find the distance between the local maximum and local minimum of the curve.

If $$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable function such that $$|f(x)| \leq 1$$ and $$f'(x)=g(x)$$, where,$$f^{2}(0)+g^{2}(0)=9$$ then prove that there exists some $$c \in(-3,3)$$ such that $$g(c) \circ g^{n}(c) < 0$$.