if f'(x)>0, then is it necessary for f(x) to be one one?ALSO,f(x)=2x+sin x .please tell if it's one-one or onto.please reply as soon as possible.......

If x₁ /= x₂ , then by Lagrange's Mean Value Theorem, we have f(x₂) - f(x₁)/(x₂-x₁) = f'(c) for some c in the interval (x₁ , x₂).

Since f'(c)>0, and x₂ > x₁ , this implies f(x₂) > f(x₁). Thus, we have that if x₁ /= x₂ , then f(x₂) /= f(x₁).

Thus f'(x)>0 proves that f(x) is one-one

In general if f(x) is monotonic, it is also one-one.