How to prove whether following inequality holds

Question

Assuming $a, b \in \mathbb{N}$, how to prove or disprove this inequality? $$ 2a \left \lceil \frac{a}{b} \right \rceil - b {\left \lceil \frac{a}{b} \right \rceil}^2 \geq \frac{a^2}{b} $$

Thanks a lot!

Answer 1

Actually for any $x\in\mathbb{R}$: $$2ax - bx^2 \le \frac{a^2}{b}$$ $$2abx - (bx)^2 \le a^2$$ $$0 \le a^2 - 2abx + (bx)^2 = (a - bx)^2$$