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.

Let a,b be integers and f(x) be a polynomial with integer coefficients such that f(b)-f(a)=1. Then, the value of b-a, is

If f(x) is a polynomial of degree at least two with integral co-efficients then the remainder when it is divided by(x-a) (x-b) is , where a ne b (1) x[(f(a)-f(b))/(b-a)] + (af(b) -bf(a))/(a-b) (2) x[(f(a) -f(b))/(a-b)] + (af(b) -bf(a))/(a-b) (3) x[(f(b) -f(a))/(a-b)] + (af(b) -bf(a))/(a-b) (4) x [ (f(b) -f(a))/(a-b)] + (bf(a) -af(a))/(a-b)

Let f(x)=a+b|x|+c|x|^(2) , where a,b,c are real constants. The, f'(0) exists if

Let f (x) be a polynomial function of degree 3 where a lt b lt c and f (a) =f (b) = f(c ). If the graph of f (x) is as shown, which of the following statements are INCORRECT ? (Where c gt|a|)

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 < d e, then the minimum number of zeroes of g(x) = f'(x)^2+f''(x)f(x) in the interval [a, e] is

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

For all real numbers x, let the mapping f (x) = 1/(x-i) . where i =sqrt -1 . If there exist real number a, b, c and d for which f(a), f(b), f(c) and f(d) form a square on the complex plane. Find the area of the square.

The figure shows the variation of photocurrent with anode potential for a photosensitve surface for three different radiations. Let l_a, l_b and l_c be the curves a, b and c, respectively (a) f_a = f_b and l_a != l_b (b) f_a = f_c and l_a = l_c (c ) f_a = f_b and l_a = l_b (d) f_b = f_c and l_b = l_c

If f"(x) exists for all points in [a,b] and (f(c )-f(a))/(c-a)=(f(b)-f( c))/(b-c),"where"a lt clt b, then show that there exists a number 'k' such that f"(k)=0.

If f(x)a n dg(x) are continuous functions in [a , b] and are differentiable in (a , b) then prove that there exists at least one c in (a , b) for which. |f(a)f(b)g(a)g(b)|=(b-a)|f(a)f^(prime)(c)g(a)g^(prime)(c)|,w h e r ea