Statement 1: If `f(0)=0,f^(prime)(x)=ln(x+sqrt(1+x^2)),` then `f(x)` is positive for all `x in R_0dot` Statement 2: `f(x)` is increasing for `x >0` and decreasing for `x<0`

If f(x)=2x+cos^(-1) x+log (sqrt(1+x^(2))-x) , then f(x) is-

Statement 1: Let f(x)=sin(cos x) \ i n \ [0,pi/2] . Then f(x) is decreasing in [0,pi/2] Statement 2: cosx is a decreasing function AAx in [0,pi/2]

If f(x) = log_e ((1-x)/(1+x)) , then f' (0) is

If f(x)= (1)/(x+1) - log (1+x), x gt 0 then f(x) is-

If f(x) = sqrt(1-sqrt1-x^2 , then at x = 0

If f(x)=2x+cot^(-1)x+log(sqrt(1+x^2)-x) then f(x) (a)increase in (0,oo) (b)decrease in [0,oo] (c)neither increases nor decreases in [0,oo] (d)increase in (-oo,oo)

Let f(x)= { x e^(a x),x lt=0;x+a x^2-x^3,x >0 where a is a positive constant. Find the interval in which f^(prime)(x) is increasing.

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real xa n dydot If f(x) is differentiable and f^(prime)(0) exists for all real permissible value of a and is equal to sqrt(5a-1-a^2)dot Then (a) f(x) is positive for all real x (b) f(x) is negative for all real x (c) f(x)=0 has real roots (d)Nothing can be said about the sign of f(x)

If f(x)=(x-1)^2-x+1 , then find f(0) .

Statement - I: if lim_(x to 0)((sinx)/(x)+f(x)) does not exist, then lim_(x to 0)f(x) does not exist. Statement - II: lim_( x to 0)(sinx)/(x)=1