Inequality #1

Algebra Level 3

If a 2 + b 2 + c 2 = 1 a^{2} + b^{2 }+ c^{2}= 1 , find the maximum value of ( a + 2 b + 3 c ) 2 (a + 2b + 3c)^{2} .


The answer is 14.

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Wage Mareto by Wage Mareto , 17, Indonesia

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1 solution

Wage Mareto
Dec 27, 2017

Relevant wiki: Applying the Cauchy Schwarz Inequality

From the Cauchy-Schwarz Inequality we have ( a 2 + b 2 + c 2 ) ( 1 2 + 2 2 + 3 2 ) ≥ ( a + 2 b + 3 c ) 2 (a^{2}+b^{2}+c^{2})(1^{2}+2^{2}+3^{2})\geq(a+2b+3c)^{2} if we simplify the equation we have ( a + 2 b + 3 c ) 2 ≤ ( 1 ) ( 14 ) = 14 (a+2b+3c)^{2}\leq(1)(14)=14

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