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^(prime)(x)=3x^2-2/(x^3) and f(1)=0 , find f(x) .

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]

Statement-1 f(x)=(sin x)/(x) lt 1 for 0 lt x lt pi/2 Statement -2 f(x)=(sin x)/(x) is decreasing function on (0,pi//2)

Let f(x)=int x^2/((1+x^2)(1+sqrt(1+x^2)))dx and f(0)=0 then f(1) is

Statement-1: If |f(x)| le |x| for all x in R then |f(x)| is continuous at 0. Statement-2: If f(x) is continuous then |f(x)| is also continuous.

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 f(x) is positive for all real x f(x) is negative for all real x f(x)=0 has real roots Nothing can be said about the sign of f(x)

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 be the function f(x)=cosx-(1-(x^2)/2)dot Then (a) f(x) is an increasing function in (0,oo) (b) f(x) is a decreasing function in (-oo,oo) (c) f(x) is an increasing function in (-oo,oo) (d) f(x) is a decreasing function in (-oo,0)

Statement-1 : f : R rarr R, f(x) = x^(2) log(|x| + 1) is an into function. and Statement-2 : f(x) = x^(2) log(|x|+1) is a continuous even function.

If f(x)=(sin^(- 1)x)^2*cos(1/ x)"if"x!=0;f(0)=0,f(x) is continuous at x= 0 or not ?