Check the nature of the following differentiable functions (i) f(x) =`e^(x)`+sin xc,x in `R^(+)` (ii)f(x)=sinx+tan x-`2x,x in(0,pi//2)`

Check the nature of the following function. (i) f(x)=sin x, x in R " (ii) " f(x) =sin x, x in N

Check the continuity of the following functions : f(x)=x^(2)" at "x=0 .

If f is a differentiable function and 2f(sinx) + f(cosx)=x AA x in R then f'(x)=

Prove that the following functions are increasing for the given intervals, (i) y =e^(x) + sinx, x epsilon R^(+) (ii) y=sin x+ tan x- 2x, x epsilon(0, pi/2) (iii) y = x + sin x , x epsilon R .

Integrate the following function w.r.t.x (i)f(x)=sinxcos3x (ii) f(x)=sin^(3)x (iii)f(x)=(sinx)/(sin(x-alpha))

Verify Rolle's theorem for each of the following functions: (i) f(x) = sin 2x " in " [0, (pi)/(2)] (ii) f(x) = (sin x + cos x) " in " [0, (pi)/(2)] (iii) f(x) = cos 2 (x - (pi)/(4)) " in " [0, (pi)/(2)] (iv) f(x) = (sin x - sin 2x) " in " [0, pi]

Separate the intervals of concavity of the following functions (i) f(x)= sin^(-1)x ,(ii) f(x)=x+sinx

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