Let R be the set of real numbers and `f : R to R` be such that for all x and y in R, `|f(x) -f(y)|^(2) le (x-y)^(3)`. Prove that f(x) is a constant.

(i) if f ( (x + 2y)/3)]= (f(x) +2f(y))/3 , " x, y in R and f (0) exists and is finite ,show that f(x) is continuous on the entire number line. (ii) Let f : R to R satisfying f(x) -f (y) | le | x-y|^(3) , AA x, y in R , The prove that f(x) is a constant function.

Let a differentiable function f:RtoR be such that for all x and y in R 2|f(x)-f(y)|le|x-y| and f^(')(x)ge(1)/(2) . So then the number of points of intersection of the graph y=f(x) with

Let f : R to R be a function such that f(0)=1 and for any x,y in R, f(xy+1)=f(x)f(y)-f(y)-x+2. Then f is

Let R be the set of all real numbers. The function f:Rrarr R defined by f(x)=x^(3)-3x^(2)+6x-5 is

Let f:R to R such that f(x+y)+f(x-y)=2f(x)f(y) for all x,y in R . Then,

Let F:R to R be such that F for all x in R (2^(1+x)+2^(1-x)), F(x) and (3^(x)+3^(-x)) are in A.P., then the minimum value of F(x) is:

Let f: R->R whre R^+ is the set of all positive real numbers, be such that f(x)=(log)_e xdot Determine: {x :f(x)=-2}

Statement 1: Let f: R -> R be a real-valued function AAx ,y in R such that |f(x)-f(y)|<=|x-y|^3 . Then f(x) is a constant function. Statement 2: If the derivative of the function w.r.t. x is zero, then function is constant.

Let f and g be real - valued functions such that f(x+y)+f(x-y)=2f(x)*g(y), forall " x , y " in R. Prove that , if f(x) is not identically zero and abs(f(x)) le 1, forall x in R, then abs(g(y)) le 1, forall y in R.

Statement 1: Let f: RvecR be a real-valued function AAx ,y in R such that |f(x)-f(y)|<=|x-y|^3 . Then f(x) is a constant function. Statement 2: If the derivative of the function w.r.t. x is zero, then function is constant.