if `f(x) =(2013) x^(2012) -(2012)x^(2011)-2014x+1007` then show that for `x in [0,1007^(1//2011)], f(x) =0` has at least one real root.

If f(x)=a x^2+b x+c ,g(x)=-a x^2+b x+c ,where ac !=0, then prove that f(x)g(x)=0 has at least two real roots.

If f(x)=int(x^8+4)/(x^4-2x^2+2)dx and f(0)=0,t h e n (a) f(x) is an odd function (b) f(x) has range R (c) f(x) has at least one real root (d) f(x) is a monotonic function.

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) .

f(x) satisfies the relation f(x)-lamda int_(0)^(pi//2)sinxcostf(t)dt=sinx If f(x)=2 has the least one real root, then

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

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)

If f(x)=Pi_(k=1)^(999)(x^(2)-47x+k) . then product of all real roots of f(x)=0 is

If f(x)=(x-1)/(x+1),x!=-1, then show that f(f(x))=-1/x provided that x != 0,-1

If f(x)=(x-1)/(x+1),x!=-1, then show that f(f(x))=-1/x provided that x!=0,1.

Let f(x)a n dg(x) be differentiable functions such that f^(prime)(x)g(x)!=f(x)g^(prime)(x) for any real xdot Show that between any two real solution of f(x)=0, there is at least one real solution of g(x)=0.