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`

If f (x) = |(0,x-a,x-b),(x+a,0,x-c),(x+b,x+c,0)| then

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]|

Let f(x) :R->R,f(x)={(2x+alpha^2,xge2),((alpha x)/2+10,x lt 2)) If f(x) is onto function then alpha belongs to (A) [1,4] (B) [-2,3] (C) [0,3] (D) [2,5]

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

Let the function f: RvecR be defined by f(x)=x+sinxforx in Rdot Then f is one-to-one and onto one-to-one but not onto onto but not-one-to-one neither one-to-one nor onto

If f: RvecR ,g(x)=f(x)+3x-1, then the least value of function y=g(|x|) is a. -9/4 b. -5/4 c. -2 d. -1

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.

Let f: R-> R , be defined as f(x) = e^(x^2)+ cos x , then is (a) one-one and onto (b) one-one and into (c) many-one and onto (d) many-one and into

Let f(x)=x+2|x+1|+2|x-1|dot If f(x)=k has exactly one real solution, then the value of k is (a) 3 (b) 0 (c) 1 (d) 2