Let f: \mathbb{R} \rightarrow \mathbb{R} be a function such that
20(x-y) \leq f(x) -f(y) \leq 20(x-y) + 2(x-y)^2
for all x,y \in \mathbb{R} \text{ and } f(0)=2 . Then f(101) equals ________________________.
Putting y=x-1 , we have,
\begin{align*}
&20 \leq f(x) -f(x-1) \leq 22 \\
\implies & \sum_{x=1}^{101} 20 \leq \sum_{x=1}^{101} f(x) - \sum_{x=1}^{101}f(x-1) \leq \sum_{x=1}^{101}22 \\
\implies & 2,000 \leq \sum_{x=1}^{101} f(x) - \sum_{x=0}^{100} f(x) \leq 2,200 \\
\implies & 2,020 \leq f(101) - f(0) \leq 2,222 \\
\implies & 2,022 \leq f(101) \leq 2,224
\end{align*}