Let `f: RvecR ,f(x)=(x-a)/((x-b)(x-c)),b > cdot` If `f` is onto, then prove that `a in (b , c)dot`

Let f(x)=(x-a)(x-b)(x-c), a lt b lt c. Show that f'(x)=0 has two roots one belonging to (a, b) and other belonging to (b, c).

Let f=NtoN be defined by f(x)=x^(2)+x+1,x inN , then prove that f is one-one but not onto.

Let f(x)=(x-a)^2+(x-b)^2+(x-c)^2dot Then, f(x) has a minimum at x= (a+b+c)/3 (b) 1/2 (c) 1/8 (d) none of these

If y=f(x)=(a x-b)/(b x-a) , show that x=f(y)dot

Let the function f: R-{-b}->R-{1} be defined by f(x)=(x+a)/(x+b) , a!=b , then (a) f is one-one but not onto (b) f is onto but not one-one (c) f is both one-one and onto (d) none of these

Let A=R-{3} and B=R-[1]dot Consider the function f: AvecB defined by f(x)=((x-2)/(x-3))dot Show that f is one-one and onto and hence find f^(-1)

Let A=R-{3} and B=R-[1]dot Consider the function f: AvecB defined by f(x)=((x-2)/(x-3))dot Show that f is one-one and onto and hence find f^(-1)

If f(x)=(x-1)/(x+1) , then prove that: (f(b)-f(a))/(1+f(b)*f(a))=(b-a)/(1+ab)

Let f be a continuous function on [a , b]dot If F(x)=(int_a^xf(t)dt-int_x^bf(t)dt)(2x-(a+b)), then prove that there exist some c in (a , b) such that int_a^cf(t)dt-int_c^bf(t)dt=f(c)(a+b-2c)dot

If g(x)=(f(x))/((x-a)(x-b)(x-c)) ,where f(x) is a polynomial of degree <3 , then intg(x)dx=|[1,a,f(a)log|x-a|],[1,b,f(b)log|x-b|],[1,c,f(c)log|x-c|]|-:|[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|+k (dg(x))/(dx)=|[1,a,-f(a)(x-a)^(-2)],[1,b,-f(b)(x-b)^(-2)],[1,c,-f(c)(x-c)^(-2)]|:-|[1,a,a^2],[1,b,b^2],[1,c,c^2]|