Prove the each of the following function is differentiable at x=0.
`(i) f(x) = cos|x| (ii) f(x) = x|x|`
`(iii) f(x)=|x^(3)|(iv) f(x) =(x)/(1+|x|)`

Which of the function is non-differential at x=0? f(x)="cos"|x|

Let A={x:-1 le x le 1}=B be a function f: A to B. Then find the nature of each of the following functions. (i) f(x) = |x| " (ii) " f(x)=x|x| (iii) f(x)=x^(3) " (iv) " f(x)="sin"(pi x)/(2)

If f(x) is a real valued function, then which of the following is injective function ? (i) f(x) = x^(5) (ii) f(x) = x + sin x (iii) f(x) = x^(2) (iv) f(x) = e^(x) (v) f(x) = x^(3) + x^(2) + 4x + 4

Find the range of the following (i) f(x) = x^(2) (ii) f(x) = x (iii) f(x) = x^(2) + 2 (iv) f(x) = x^(3)

Find the range of the following functions given by f(x)=(3)/(2-x^(2)) " " (ii) f(x)=1-|x-2| (iii) f(x)=|x-3| " " (iv) f(x)=1+3 cos 2x

Find the domain and range of the following functions. (i) f(x)=sqrt(2x-3) " (ii) " f(x) =(1)/(x-2) (iii) f(x) =x^(2) +3 " (iv) " f(x)=(1)/(x^(2)+2)

Find the domain of each of the following functions given by f(x)=1/(sqrt(x-|x|)) (ii) f(x)=1/(sqrt(x+|x|)) (iii) f(x)=1/(sqrt(x-[x])) (iv) f(x)=1/(sqrt(x+[x]))

Find the domain of each of the following real valued functions of real variable: f(x)=sqrt(x-2) (ii) f(x)=1/(sqrt(x^2-1)) (iii) f(x)=sqrt(9-x^2) (iv) f(x)=sqrt((x-2)/(3-x))

Find the range of each of the following functions: f(x)=1/(sqrt(x-5)) (ii) f(x)=sqrt(6-x^2) (iii) f(x)=x/(1-x^2) (iv) f(x)=3/(2-x^2)

Find the intervals of monotonicity of the following functions. (i) f(x) =-x^(3) +6x^(2)-gx-2 (ii) f(x) =x+(1)/(x+1) (iii) f(x) =x. e^(x-x^(2)) (iv) f(x) =x- cosx