Question Number 142325 by loveineq last updated on 30/May/21
![Given that a ≥ 1 ≥ b > 0. Prove the followings: (1) (1/2)(a−b)^2 ≤ (a−1)^2 +(1−b)^2 ≤ (a−b)^2 (2) (1/4)(a−b)^3 ≤ (a−1)^3 +(1−b)^3 ≤ (a−b)^3](https://www.tinkutara.com/question/Q142325.png)
$$\mathrm{Given}\:\mathrm{that}\:{a}\:\geqslant\:\mathrm{1}\:\geqslant\:{b}\:>\:\mathrm{0}.\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{followings}:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\left({a}−{b}\right)^{\mathrm{2}} \:\leqslant\:\left({a}−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{1}−{b}\right)^{\mathrm{2}} \:\leqslant\:\left({a}−{b}\right)^{\mathrm{2}} \:\:\:\: \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{4}}\left({a}−{b}\right)^{\mathrm{3}} \:\leqslant\:\left({a}−\mathrm{1}\right)^{\mathrm{3}} +\left(\mathrm{1}−{b}\right)^{\mathrm{3}} \:\leqslant\:\left({a}−{b}\right)^{\mathrm{3}} \\ $$$$ \\ $$