Let $()$ be continuous on $[0,∞)$ and twice differentiable on $(0,∞)$

Question

Let $()$ be continuous on $[0,∞)$ and twice differentiable on $(0,∞)$. If $′()> 0$ and $″()<0$ on $[1,∞)$ prove that for all $ ≠ ≥ 1$, $|()−()|<′(1)|−|$.

Hello, can anyone help me with this question? Thank you!

Answer 1

Simple. Use the MVT which says: $f(x) - f(y) = f'(c)(x-y)\implies |f(x)-f(y)| = |f'(c)||x-y|=f'(c)|x-y|< f'(1)|x-y| $ since $f'(c) > 0$, and $f'(c) < f'(1)$ since $c > x \ge 1$ as $f'(x)$ is a decreasing function.