Show that there exists no polynomial `f(x)` with integral coefficients which satisfy `f(a)=b ,f(b)=c ,f(c)=a ,` where `a , b , c ,` are distinct integers.

The polynomial f(x)=x^4+a x^3+b x^3+c x+d has real coefficients and f(2i)=f(2+i)=0. Find the value of (a+b+c+d)dot

Let f(x) be a polynomial function of second degree. If f (1) =f(-1) and a,b,c are in A.P. then f (a) , f'(b) , f'(c ) are in-

The domain of f(x)=ln(a x^3+(a+b)x^2+(b+c)x+c), where a >0, b^2-4ac=0 , is

The function f satisfying (f(b)-f(a))/(b-a)nef'(x) for any "x"in(a,b) is

Let f(x) = x^4 + ax^3 + bx^2 + cx + d be a polynomial with real coefficients and real roots. If |f(i)|=1where i=sqrt(-1) , then the value of a +b+c+d is

If a polynomial function f(x) satisfies f(f(f(x))=8x+21 , where p and q are real numbers, then p+q is equal to _______

If a polynomial function f(x) satisfies f(f(f(x))=8x+21 , where p and q are real numbers, then p+q is equal to _______

If f(x) is a twice differentiable function such that f(a)=0, f(b)=2, f(c)=-1,f(d)=2, f(e)=0 where a < b < c < de, then the minimum number of zeroes of g(x) = f'(x)^2+f''(x)f(x) in the interval [a, e] is

STATEMENT 1: If f(x) is continuous on [a , b] , then there exists a point c in (a , b) such that int_a^bf(x)dx=f(c)(b-a) STATEMENT 2: For a < b , if ma n dM are, respectively, the smallest and greatest values of f(x)on[a , b] , then m(b-a)lt=int_a^bf(x)dxlt=(b-a)Mdot

Let f(x) be continuous functions f: RvecR satisfying f(0)=1a n df(2x)-f(x)=xdot Then the value of f(3) is 2 b. 3 c. 4 d. 5