Is a function with these conditions both convex and concave?

Question

Let $f : [a,b] \to \mathbb{R}, f‘(x)>0, f‘‘(x)<=0, f(a)<0<f(b)$, so f is strictly monotonous increasing and concave. Also let f satisfy $b-\frac{f(b)}{f‘(b)} >= a$.

Is f a straight line? (My goal is to show that the newton method converges to the unique root of f)

Answer 1

No. Take $f(x)=\log(x+1)$, $a=\frac{-1}{2}$ and $b=\frac{1}{2}$.