Continuity And Differentiability

Let f (x) = min { x, } for every real number of x, then

Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit

is equal to a nonzero real number, is_____

For a polynomial with real coefficients, let denote the number of distinct real roots of . Suppose in the set of polynomials with real coefficients defined by

For a polynomial , let and denote its first and second order derivatives, respectively. Then the minimum possible value of , where , is

Let be the function defined by
If are such that , then the value of is ______