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

Let R denote the set of all real numbers. For a real number x, let [x] denote the greatest integer less than or equal to x. Let n denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option.

1. (P) → (1), (Q) → (3), (R) → (2), (S) → (5)
2. (P) → (2), (Q) → (1), (R) → (4), (S) → (3)
3. (P) → (5), (Q) → (1), (R) → (4), (S) → (3)
4. (P) → (2), (Q) → (3), (R) → (1), (S) → (5)

Let R denote the set of all real numbers. Let f: R R be defined by f(x)= cases 6x+ x2x+ x & if x 0,\\[2mm] 73 & if x=0. cases Then which of the following statements is (are) TRUE?

1. The point x=0 is a point of local maxima of f
2. The point x=0 is a point of local minima of f
3. Number of points of local maxima of f in the interval [ ,6 ] is 3
4. Number of points of local minima of f in the interval [2 ,4 ] is 1

Let the function f: R R be defined by f(x)= xe x (x2023+2024x+2025 ) (x2-x+3 )+ 2e x (x2023+2024x+2025 ) (x2-x+3 ). Then the number of solutions of f(x)=0 in R is .