Prove inequality: $74 - 37\sqrt 2 \le a+b+6(c+d) \le 74 +37\sqrt 2$ without calculus 
Let $a,b,c,d \in \mathbb R$ such that $a^2 + b^2 + 1 = 2(a+b),
c^2 + d^2 + 6^2 = 12(c+d)$, prove inequality without calculus (or
  langrange multiplier):  $$74 - 37\sqrt 2  \le a+b+6(c+d) \le 74
 +37\sqrt 2$$

The original problem is find max and min of  $a+b+6(c+d)$ where ...
Using some calculus, I found it, but could you solve it without calculus.

Hint: You can split this problem to find max and min of $a+b$ and $c+d$.
