Let `f(x)=Ax^2+Bx+C` where A, B, C are three real constants, if f(x) is integer for integral values of x, then prove that each of 2A, (A+B) and C is an integer. Conversely, if each of 2A, (A+B) , C is an integer then f(x) will be integer for integral values of x.

Let f(x) = Ax^2+Bx+C where A,B,C are real numbers. Prove that if f(x) is an integer whenever x is an integer, then the number 2A, A+B and C are all integers. Conversely, prove that if the numbers 2A, A+B and C are all integer then f(x) is an integer whenever x is an integer.

Let f(x)=A x^2+B x+c ,w h e r eA ,B ,C are real numbers. Prove that if f(x) is an integer whenever x is an integer, then the numbers 2A ,A+B ,a n dC are all integer. Conversely, prove that if the number 2A ,A+B ,a n dC are all integers, then f(x) is an integer whenever x is integer.

Let f(x) = a+ b|x| + c|x|^4 where a, b and continuous are real constants . Then, f(x) is differentiable at x = 0, if

If f(x)={{:("0, when x is an integer ,"),("2, when x is not an integer,"):} then f(pi) =

Let f(x)=a x^2+bx +c a ,b ,c in R . If f(x) takes real values for real values of x and non-real values for non-real values of x , then (a) a=0 (b) b=0 (c) c=0 (d) nothing can be said about a ,b ,c .

In a A B C ,ifA B=x , B C=x+1,/_C=pi/3 , then the least integer value of x is

If f(x) = 0 when x is an integer. = 2 when x is not an integer, then find - (a) f(0), (b) f(sqrt2) .

If f(x)= x^(3) + ax^(2) + bx + c is an increasing function for all real values of x then-

Let f(x)={1+|x|,x<-1 [x],xgeq-1 , where [.] denotes the greatest integer function. The find the value of f{f(-2.3)}dot

If f(x)=(ax-b)/(bx-a) , then find the value of f(1/x) .