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.

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) .

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.

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

If f(x) be a quadratic polynomial such that f(x)=0 has a root 3 and f(2)+f(-1)=0 then other root lies in

Let f(x) be a polynomial of degree 5 such that f(|x|)=0 has 8 real distinct , Then number of real 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

If f(x) is a polynomial of degree 5 with real coefficients such that f(|x|)=0 has 8 real roots, then f(x)=0 has

If f(x)=x^(3)-3x+1, then the number of distinct real roots of the equation f(f(x))=0 is

Let a in R and f : R rarr R be given by f(x)=x^(5)-5x+a , then (a) f(x)=0 has three real roots if a gt 4 (b) f(x)=0 has only one real root if a gt 4 (c) f(x)=0 has three real roots if a lt -4 (d) f(x)=0 has three real roots if -4 lt a lt 4