Let `F:R- R` be a thrice differentiable function. Suppose that `F(1)=0,F(3)=-4` and `F(x)<0` for all `x in (1/2,3)`. Let `f(x)=xF(x)` for all `x in R` The correct statement is

Let F : R to R be a thrice differentiable function . Suppose that F(1)=0,F(3)=-4 and F(x) lt 0 " for all" x in (1,3). f(x) = x F(x) for all x inR . The correct statement (s) is / are

Let F : R to R be a thrice differentiable function . Suppose that F(1)=0,F(3)=-4 and F(x) lt 0 " for all" x in (1/2,3). f(x) = x F(x) for all x inR . The correct statement (s) is / are

Let F : R to R be a thrice differentiable function . Suppose that F(1)=0,F(3)=-4 and F(x) lt 0 " for all" x in (1,3). f(x) = x F(x) for all x inR . Ifint_(1)^(3)x^(2)F'(x)dx=-12 andint_(1)^(3)x^(3)F'' (x)dx =40 , then the correct expression (s) is //are

Let F : R to R be a thrice differentiable function . Suppose that F(1)=0,F(3)=-4 and F(x) lt 0 for all x in (1,3) . f(x) = x F(x) for all x inR . If int_(1)^(3)x^(2)F'(x)dx=-12 andint_(1)^(3)x^(3)F'' (x)dx =40 , then the correct expression (s) is //are

Let F:RR to RR be a thrice differentiable function. Suppose that F(1)=0,F(3)=-4 and F'(x)lt0 for all x in((1)/(2),3) . Let f(x)=xF(x) for all x in RR . The correct statement(s) is (are)-

Let F:RrarrR be a thrice differentiable function. Suppose that F(1)=0, F(3)=-4 and F'(x)lt0 for all x in (1//2,3) . Let f(x)=xF(x) for all x in R . If int_1^3x^2F'(x)dx=-12 and int_1^3x^3F''(x)dx=40 , then the correct expression(s) is (are)

Let F : R rarr R be a thrice differentiable function, Supose that F(1) = 0, F(3) = -4 and F'(x) lt 0 for all x in (1//2//3) . Let f(x) = xF(x) for all x in R . If underset(1)overset(3)int_(1)^(3) x^(2)F'(x) dx = -12 and int_(1)^(3)x^(3)F"(x)dx = 40 . Then the correct expression(s) is are :

Let f:R rarr R be a twice differentiable function such that f(x+pi)=f(x) and f'(x)+f(x)>=0 for all x in R. show that f(x)>=0 for all x in R .

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f '(x) ne 0 for all x in R . If |[f(x)" "f'(x)], [f'(x)" "f''(x)]|= 0 , for all x in R , then the value of f(1) lies in the interval:

Let f : R to R be a differentiable function such that f(2) = -40 ,f^1(2) =- 5 then lim_(x to 0)((f(2 - x^2))/(f(2)))^(4/(x^2)) is equal to