Question

If $a^2+b^2+c^2+d^2=25$, then out of the following statements which one is correct?

• A. $ab+bc+cd+da\le 25$
• B. $ab+bc+cd+da\ge 25$
• C. $ab+bc+cd+da\le \frac{25}{2}$
• D. $ab+bc+cd+da\ge \frac{25}{2}$

Answer 1

The correct option is A $ab+bc+cd+da\le 25$

Consider ${(a-b)}^2+{(b-c)}^2+{(c-d)}^2+{(d-a)}^2\ge 0$

$2{(a}^2+b^2+c^2+d^2)-2(ab+bc+cd+da)\ge 0$

$⟹{(a}^2+b^2+c^2+d^2)\ge (ab+bc+cd+da)$

$⟹25\ge (ab+bc+cd+da)$ ...[given ${(a}^2+b^2+c^2+d^2=25\left)\right]$.

Hence proved.