How can function mix with inequality

Algebra Level 5

If g ( x ) g(x) is a polynomial with

x 2 + 2 x + 666 ≤ g ( x ) ≤ 2 x 2 + 4 x + 667 x^2+2x+666 \le g(x) \le 2x^2+4x+667 for all real x x and g ( 1 ) = 671 g (1) =671 ,

find g ( 3 ) g(3) .


The answer is 689.

Choi Chakfung by choi chakfung , 28, Hong Kong

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2 solutions

Choi Chakfung
May 13, 2016

r e w r i t e t h e e q u a t i o n , ( x + 1 ) 2 + 665 ≤ g ( x ) ≤ 2 ( x + 1 ) 2 + 665 ⇒ g ( x ) = a ( x + 1 ) 2 + 665 w h e r e 1 ≤ a ≤ 2 g ( 1 ) = 671 671 = a ( 1 + 1 ) 2 + 665 ⇒ a = 1.5 g ( 3 ) = 1.5 ( 3 + 1 ) 2 + 665 = 689 rewrite\quad the\quad equation\quad ,\\ { (x+1) }^{ 2 }+665\le g(x)\le 2({ x+1) }^{ 2 }+665\Rightarrow g(x)=a{ (x+1) }^{ 2 }+665\quad where\quad 1\le a\le 2\\ g(1)=671\\ 671=a{ (1+1) }^{ 2 }+665\Rightarrow a=1.5\\ g(3)=1.5{ (3+1) }^{ 2 }+665=689

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