The point(s) in the curve $${y^3} + 3{x^2} = 12y$$ where the tangent is vertical, is (are)

The length of a longest interval in which the function $$3\,\sin x - 4{\sin ^3}x$$ is increasing, is

Let $$f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$$ and let $$m(b)$$ be the minimum value of $$f(x)$$. As $$b$$ varies, the range of $$m(b)$$ is

The triangle formed by the tangent to the curve $$f\left( x \right) = {x^2} + bx - b$$ at the point $$(1, 1)$$ and the coordinate axex, lies in the first quadrant. If its area is $$2$$, then the value of $$b$$ is