If `a and b` are two distinct real roots of the polynomial `f(x)` such that `a < b`, then there exists a real number c lying between `a and b`, such that

Let a and b be two distinct roots of a polynomial equation f(x) =0 Then there exist at least one root lying between a and b of the polynomial equation

Let a and b be two distinct roots of a polynomial equation f(x) =0 Then there exist at least one root lying between a and b of the polynomial equation

Suppose a , b denote the distinct real roots of the quadrtic polynomial x^(2)+20x-2020 and suppose c,d denote the distinct complex roots of the quadratic polynomial x^(2)-20x+2020 , then the value of ac(a-c)+ad(a-d) bc(b-c)bd(b-d) is

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

Let p(x)= x^2+ax+b have two distinct real roots, where a, b are real numbers. Define g(x)= p(x^3) for all real numbers x. Then which of the following statements are true? I. g has exactly two distinct real roots. II. g can have more than two distinct real roots III. There exists a real number alpha such that g(x) ge alpha for all real x

Let f(x)=0 be a polynomial equation with real coefficients. Then between any two distinct real roots of f(x)=0 , there exists at least one real root of the equation f'(x)=0 . This result is a consequence of the celebrated Rolle's theorem applied to polynomials. Much information can be extracted about the roots of f(x)=0 from the roots of f'(x)=0 . The range of values of k for which the equation x^(4)-14x^(2)+24x-k=90 has four unequal real roots is

Let f(x)=0 be a polynomial equation with real coefficients. Then between any two distinct real roots of f(x)=0 , there exists at least one real root of the equation f'(x)=0 . This result is a consequence of the celebrated Rolle's theorem applied to polynomials. Much information can be extracted about the roots of f(x)=0 from the roots of f'(x)=0 . The range of values of k for which the equation x^(4)+4x^(3)-8x^(2)+k=0 has four real and unequal roots is

Let f(x)=0 be a polynomial equation with real coefficients. Then between any two distinct real roots of f(x)=0 ,there exists at least one real root of the equation f'(x)=0 . This result is a consequence of the celebrated Rolle's theorem applied to polynomials. Much information can be extracted about the roots of f(x)=0 from the roots of f'(x)=0 . Q.The exhaustive range of values of k for which the equation x^(4)-14x^(2)+24x-k=0 has four unequal real roots is