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.

Let f(x) be a polynomial with integral coefficients. If f(1) and f(2) both are odd integers, prove that f(x) = 0 can' t have any integral root.

Let f(x) be a quadratic polynomial with integer coefficients such that f(0) and f(1) are odd integers.then prove that the equation f(x)=0 does not have an integer solution

If f'(x)>0,f(x)>0AA x in(0,1) and f(0)=0,f(1)=1, then prove that 'f(x)f^(^^)-1(x)

Let f(x)=x^2-bx+c,b is an odd positive integer. Given that f(x)=0 has two prime numbers as roots and b+c=35. If the least value of f(x) AA x in R" is "lambda," then "[|lambda/3|] is equal to (where [.] denotes greatest integer function)

If f(x) is a real valued polynomial and f (x) = 0 has real and distinct roots, show that the (f'(x)^(2)-f(x)f''(x))=0 can not have real roots.

If f(x)=(a-x^(n))^((1)/(n)),a>0 and n in N, then prove that f(f(x))=x for all x.

f(x) is a cubic polynomial x^(3)+ax^(2)+bx+c such that f(x)=0 has three distinct integral roots and f(g(x))=0 does not have real roots,where g(x)=x^(2)+2x-5, the minimum value of a+b+c is

If f((x + 2y)/3)=(f(x) + 2f(y))/3 AA x, y in R and f(x) is continuous at x = 0. Prove that f(x) is continuous for all x in R.

Let f(x)=ax^(2)+bx+c, where a,b,c,in R and a!=0, If f(2)=3f(1) and 3 is a roots of the equation f(x)=0, then the other roots of f(x)=0 is

Let f(x)=ax^(3)+bx^(2)+cx+d,a!=0 If x_(1) and x_(2) are the real and distinct roots of f'(x)=0 then f(x)=0 will have three real and distinct roots if