Question Number 32767 by Rasheed.Sindhi last updated on 01/Apr/18
$$\mathrm{For}\:{a},{b},{c}\geqslant\mathrm{0}\:\mathrm{if}\:{a}+{b}+{c}={n},\:\mathrm{determine} \\ $$$$\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum}\:\mathrm{values}\:\mathrm{of} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}. \\ $$
Commented by Rasheed.Sindhi last updated on 02/Apr/18
$$\mathrm{Is}\:\mathrm{it}\:\mathrm{correct}\: \\ $$$${a}+{b}+{c}\leqslant{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\leqslant\left({a}+{b}+{c}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:? \\ $$
Commented by mrW2 last updated on 05/Apr/18
$$\mathrm{0}\leqslant{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\leqslant{n}^{\mathrm{2}} \\ $$