A not-so-obvious maximum

Algebra Level 4

Given that a , b , c a,b,c are positive real numbers such that a + b + c ≤ 9 a+b+c \leq 9 , find the maximum value of

( a 2 + b 2 + c 2 ) ( 2 a b + 2 b c + 2 c a + 5 ) . \left(a^2+b^2+c^2\right)(2ab+2bc+2ca+5).


The answer is 1849.

Takeda Shigenori by Takeda Shigenori , 46, Malaysia

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2 solutions

Takeda Shigenori
Nov 16, 2017

From the AM-GM inequality,

( a 2 + b 2 + c 2 ) + ( 2 a b + 2 b c + 2 c a + 5 ) 2 ≥ ( a 2 + b 2 + c 2 ) ( 2 a b + 2 b c + 2 c a + 5 ) \frac{(a^2+b^2+c^2)+(2ab+2bc+2ca+5)}{2} \geq \sqrt{(a^2+b^2+c^2)(2ab+2bc+2ca+5)}

The left hand side of the inequality can be simplified to ( a + b + c ) 2 + 5 2 ≤ 43 \frac{(a+b+c)^2+5}{2} \leq 43 , and therefore

( a 2 + b 2 + c 2 ) ( 2 a b + 2 b c + 2 c a + 5 ) ≤ 4 3 2 = 1849 (a^2+b^2+c^2)(2ab+2bc+2ca+5) \leq 43^2=1849

An example of a solution is a = 3 − 2 2 , b = 3 , c = 3 + 2 2 a=3-2\sqrt{2},b=3,c=3+2\sqrt{2} . Note that all three values of a , b , c a,b,c can be different in order to let the expression attain the maximum value even though the expression itself is symmetric. Notice also that a = b = c = 3 a=b=c=3 , the obvious symmetrical values, is not the solution.

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Amazing. your solution is satisfying all those conditions the cruicial one is A^2+b^2+c^2=2ab+2bc+2ca+5. This is the main obstruction for thinking in this way. We are not sure if the constraints can be satisfied. Largrangre multipliers are of little use in this problem.

Srikanth Tupurani - 2 years, 7 months ago
Abhinav Shripad
Nov 26, 2017

On squaring the a+b+c=<9 We get a^2 + b^2+c^2+2x=<81 So the required is (81-2x)(2x+5) =405+152x-4x^2 On differentiating we get that the maximum value of x= 19 Then we put that in the required and get it is maximum value is 1849

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Very nice approach!

Note: You also need to verify that a b + b c + c a = 18 ab+bc+ca = 18 can be achieved subject to a + b + c ≤ 9 a + b + c \leq 9 . Otherwise, we have to find the range of x x under the restriction, and then determine when the max of 405 + 152 x − 4 x 2 405 + 152x - 4x^2 under this domain restriction.

Calvin Lin Staff - 3 years, 6 months ago

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