Consider the polynomial f`(x)= 1+2x+3x^2+4x^3`. Let s be the sum of all distinct real roots of `f(x)`and let `t= |s|`.

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

For a polynomia g(x) with real coefficients, let m_(g) denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by S={(x^(2)-1)^(2)(a_(0)+a_(1)+a_(2)x^(2)+a_(3)x^(3)): a_(0) a_(1), a_(2), a_(3)= RR} For a polynomial f. let f and f denote its first and second order derivties, respectively. Then the minimum possible value of (m_(f')+m_(f")) . where f in S is, ____

Paragraph for Question Nos. 12 to 14 Consider f(x) = x^3 * (x-2)^2 *(x-1), then 12. The number of distinct real roots of equation f(x) = f(3/2) is (A) 2 (B) 3 (C)4 (D) 6 13. The number of distinct real roots of equation f(x) = f(2/3) (D)4 (C)3 (B) 2 (A) 1

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 S be the set of all real roots of the equation 4^x (4^x -1)+2=|4^x -1|+|4^x-2) .Then S:

Let f (x)=(x+1) (x+2) (x+3)…..(x+100) and g (x) =f (x) f''(x) -f ('(x)) ^(2). Let n be the numbers of rreal roots of g(x) =0, then:

Let f(x)=(e^(x))/(x^(2))x in R-{0}. If f(x)=k has two distinct real roots,then the range of k, is