Let `f(x)=x+f(x-1)forAAx in RdotIff(0)=1,fin df(100)dot`

Let f: RvecR be a function satisfying condition f(x+y^3)=f(x)+[f(y)]^3 for all x ,y in Rdot If f^(prime)(0)geq0, find f(10)dot

If f(x) is a polynomial function satisfying f(x)dotf(1/x)=f(x)+f(1/x) and f(4)=65 ,t h e nfin df(6)dot

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then find the value of f(7)dot

Let g(x)=f(logx)+f(2-logx)a n df^(x)<0AAx in (0,3)dot Then find the interval in which g(x) increases.

Let f(x)=x^2-2x ,x in R ,a n dg(x)=f(f(x)-1)+f(5-f(x))dot Show that g(x)geq0AAx in Rdot

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then prove that int_0^2f(2-x)dx=2int_0^1f(x)dxdot

Let g(x)=2f(x/2)+f(2-x)a n df^('')(x)<0AAx in (0,2)dot Then g(x) increases in (a) (1/2,2) (b) (4/3,2) (c) (0,2) (d) (0,4/3)

Let f(x y)=f(x)f(y)AAx , y in Ra n df is differentiable at x=1 such that f^(prime)(1)=1. Also, f(1)!=0,f(2)=3. Then find f^(prime)(2)dot

Let g(x)=f(x)+f(1-x) and f "(x)>0AAx in (0,1)dot Find the intervals of increase and decrease of g(x)dot