Find Maximum and Minimum value by two polynomial equations

Question

Suppose there are $7$ real numbers say $A,B,C,D,E,F,G$ All we need to find the minimum and maximum value of $G$ satisfying the following two equations :-

Sum of Numbers :- $A + B + C + D + E + F + G = 15$

Sum of Square of Numbers :- $A^2 + B^2 + C^2 + D^2 + E^2 + F^2 + G^2 = 33$

I could not figure not how to approach this question ! If possible, somebody provide me the general solution for $N$ number of variables ?

Answer 1

Hint,use Cauchy-Schwarz inequality we have $$(A^2+B^2+C^2+D^2+E^2+F^2)(1+1+1+1+1+1)\ge (A+B+C+D+E+F)^2$$ then we have $$6(33-G^2)\ge (15-G)^2$$ then we have to find to $G$ maximum and minimum

so for the general solution of $n$ number of variables ?

you can note $$[a^2_{1}+a^2_{2}+\cdots+a^2_{n-1}][1+1+\cdots+1]\ge [a_{1}+a_{2}+\cdots+a_{n-1}]^2$$

Answer 2

Hint

Sum of numbers: Set $A= B=\ldots = E=0,F = 15-G$. What values of $G$ can you choose?

Sum of squared numbers: Note that $A^2+\ldots +F^2 \geq 0$ and try $A= B=\ldots = E=F=0$. What values of $G$ can you choose?