Maximum value of a variable in system of two equations

Consider the system of equations

$$3a + 2b + c+ d =14$$ $$a^{2} + b^{2} + c^{2} + d^{2} =14.$$

Is there any way to find maximum value of $$d$$ using AM-GM inequality. I am not even able to think, how to start approaching this problem.

• Are $a,b,c,d \ge 0$?
– vvg
Nov 21, 2022 at 12:41
• No, these are any real numbers. Nov 21, 2022 at 12:43
• Using Cauchy-Schwarz (or with AM-GM after expanding if you want): $(3a+2b+c)^2 \leq 14(a^2+b^2+c^2) \implies (14-d)^2 \leq 14(14-d^2)$. Can you finish it from here?
– LHF
Nov 21, 2022 at 12:56
• Can we say $d \le \frac{28}{15}$? Nov 21, 2022 at 13:07
• @Math__Nerd yes, that's the answer.
– LHF
Nov 21, 2022 at 13:09