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Algebra Level 3

If a , b , c , d a,b,c,d are positive real numbers , such that a + 2 b + 3 c + 4 d ≥ 30 a + 2b + 3c + 4d \ge 30 , then enter the minimum value of the following expression a 2 + b 2 + c 2 + d 2 \Large a^2 + b^2 + c^2 + d^2


The answer is 30.

Rishu Jaar by Rishu Jaar , 21, India

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1 solution

a 2 + b 2 + c 2 + d 2 = ( a 2 + 1 ) + ( b 2 + 4 ) + ( c 2 + 9 ) + ( d 2 + 16 ) − 30 a^2+b^2+c^2+d^2=(a^2+1)+(b^2+4)+(c^2+9)+(d^2+16)-30

Using AM GM inequality inside the brackets

a 2 + b 2 + c 2 + d 2 ≥ 2 a + 4 b + 6 c + 8 c − 30 a^2+b^2+c^2+d^2 \ge 2a+4b+6c+8c-30

Given that,

2 a + 4 b + 6 c + 8 c ≥ 60 2a+4b+6c+8c\ge 60

So,

a 2 + b 2 + c 2 + d 2 ≥ 60 − 30 a^2+b^2+c^2+d^2 \ge 60-30

a 2 + b 2 + c 2 + d 2 ≥ 30 a^2+b^2+c^2+d^2 \ge 30

Equality occurs when a = 1 a=1 , b = 2 b=2 , c = 3 c=3 and d = 4 d=4 .

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perfect solution !(+1)

Rishu Jaar - 3 years, 7 months ago

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