If `f'(x)>0 and f''(x)>0AAx inR,` then for any two real numbers `x_1 and x_2,(x_1!=x_2)`

If f(x)gt0 and f"(x)gt0 forallx in R, then for any two real numbers x_1 and x_2,(x_1nex_2)

If f'(x)>0,AA x in R, then for any two real in shers x_(1) and x_(2)(x_(1)!=x_(2))

If f(x)=x|x|,AAx inR . then :

If f(x)=(x)/(1+|x|)" for "x inR , then f'(0) =

If f(x)=(x)/(1+|x|)" for "x inR , then f'(0) =

Let f(x)=x/(1-x) and ' a ' be a real number. If x_0=a , x_1=f(x_0) , x_2=f(x_1) , x_3=f(x_2) and so on. If x_(2011)=- (1/2012) , then the value of reciprocal of ' a ' is

Let f_(1) (x) and f_(2) (x) be twice differentiable functions where F(x)= f_(1) (x) + f_(2) (x) and G(x) = f_(1)(x) - f_(2)(x), AA x in R, f_(1) (0) = 2 and f_(2) (0) = 1. "If" f'_(1)(x) = f_(2) (x) and f'_(2) (x) = f_(1) (x) , AA x in R . then the number of solutions of the equation (F(x))^(2) =(9x^(4))/(G(x)) is...... .