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) 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

Statement-1: If alpha and beta are real roots of the quadratic equations ax^(2) + bx + c = 0 and -ax^(2) + bx + c = 0 , then (a)/(2) x^(2) + bx + c = 0 has a real root between alpha and beta Statement-2: If f(x) is a real polynomial and x_(1), x_(2) in R such that f(x_(1)) f_(x_(2)) lt 0 , then f(x) = 0 has at leat one real root between x_(1) and x_(2) .

If f(x)=2 for all real numbers x, then f(x+2)=

Show that the equation x^2+a x-4=0 has real and distinct roots for all real values of a .

Statement-1: The equation (pi^(e))/(x-e)+(e^(pi))/(x-pi)+(pi^(pi)+e^(e))/(x-pi-e) = 0 has real roots. Statement-2: If f(x) is a polynomial and a, b are two real numbers such that f(a) f(b) lt 0 , then f(x) = 0 has an odd number of real roots between a and b.

Statement-1: If a, b, c, A, B, C are real numbers such that a lt b lt c , then f(x) = (x-a)(x-b)(x-c) -A^(2)(x-a)-B^(2)(x-b)-C^(2)(x-c) has exactly one real root. Statement-2: If f(x) is a real polynomical and x_(1), x_(2) in R such that f(x_(1)) f(x_(2)) lt 0 , then f(x) has at least one real root between x_(1) and x_(2)

Let f(x)=x^(2)+ax+b , where a, b in R . If f(x)=0 has all its roots imaginary, then the roots of f(x)+f'(x)+f''(x)=0 are

If f(x) is an even function then find the number of distinct real numbers x such that f(x)=f((x+1)/(x+2)).

If f is real-valued differentiable function such that f(x)f'(x)<0 for all real x, then

If f is real-valued differentiable function such that f(x)f'(x)<0 for all real x, then