Let a, b, c d ∈ R be such that a2+b2+c2+d2=25, then

Question

Let a, b, c d ∈ R be such that a2+b2+c2+d2=25, then (A) a b+b c+c d+a d ≤ (25/2) (B) a b+b c+c d+a d ≤ 25 (C) a b+b c+c d+a d ≤ 5 (D) ab + bc + cd + ad ≤ (5/2)

Tardigrade Admin · Accepted Answer

text Given, a 2+ b 2+ c 2+ d 2=25 text As, ( a - b )2+( b - c )2+( c - d )2+( d - a )2 ≥ 0 ⇒ 2( a 2+ b 2+ c 2+ d 2) ≥ 2( ab + bc + cd + ad ) ⇒ ab + bc + cd + da ≤ 25

Q. Let $a, b, c d \in R$ be such that $a^{2} + b^{2} + c^{2} + d^{2} = 25$ , then

$ab + b c + c d + a d \leq \frac{25}{2}$

$ab + b c + c d + a d \leq 25$

$ab + b c + c d + a d \leq 5$

ab + bc + cd + ad $\leq \frac{5}{2}$

$Given, a^{2} + b^{2} + c^{2} + d^{2} = 25$
$As, (a - b)^{2} + (b - c)^{2} + (c - d)^{2} + (d - a)^{2} \geq 0$
$\Rightarrow 2 (a^{2} + b^{2} + c^{2} + d^{2}) \geq 2 (ab + b c + c d + a d)$
$\Rightarrow ab + b c + c d + d a \leq 25$

Q. Let a,b,cd∈R be such that a2+b2+c2+d2=25, then

ab+bc+cd+ad≤225​

ab+bc+cd+ad≤25

ab+bc+cd+ad≤5

ab + bc + cd + ad ≤25​