Application of Derivatives

Consider the function f : (0, ) (- , ) given by f(x) = x e(x) - x + 1 . Then which one of the following statements is TRUE ?

1. The derivative of the function f is decreasing in the interval (0, 1)
2. The function f has a local maximum at some point a (0, )
3. The function f has a local minimum at some point b (0, )
4. The function f has NEITHER a point of local maximum NOR a point of local minimum in the interval (0, )

Let R denote the set of all real numbers. Define the function f: R R by f(x) = cases 2 - 2x2 - x2 1x & if x 0, \ 2 & if x = 0. cases Then which one of the following statements is TRUE?

1. The function f is NOT differentiable at x = 0
2. There is a positive real number , such that f is a decreasing function on the interval (0, )
3. For any positive real number , the function f is NOT an increasing function on the interval (- , 0)
4. x = 0 is a point of local minima of f