If `f(x)=x^3+b x^2+c x+d` and` f(0),f(-1)` are odd integers, prove that `f(x)=0` cannot have all integral roots.

If f(x)=x^2-3x+1 , then find f(0) .

If f(x)=(1-x)/(1+x) , then prove that f(x)+f(1/x)=0 .

If f(x)=(a-x^(n))^(1/n),agt0 and n is a positive integer, then prove that f(f(x)) = x.

If f(x+f(y))=f(x)+yAAx ,y in R and f(0)=1, then prove that int_0^2f(2-x)dx=2int_0^1f(x)dx .

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.

If f(x)= x^5+x^3-2x+3 ,Prove that f'(1)+f'(-1)=4f(0).

If f(x)=(1-x)/(1+x), prove that f{f((1)/(x))}=1/x[x ne0]

If f(x-2)=2x^2+3x-5 , find f(x) and prove that f(-1) = 0.

f(x) be a quadratic polynomial f(x) = (x-1) (ax+b) satisfying f(2) + f(4) = 0. If unity is one root of f(x) = 0 then find the other root.

If f(a-x)=f(a+x) " and " f(b-x)=f(b+x) for all real x, where a, b (a gt b gt 0) are constants, then prove that f(x) is a periodic function.