Not Only AM-GM 1 (The Other AM-GM Part 3)
Let M be the maximum and m be the minimum value of the expression x 2 + 2 x 2 + 4 x , for x is real integer (not necessary positive). Find M − m .
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The answer is 3.
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2 solutions
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Hey Ankit! How are you? Why you tagged me?
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Hi!! Zuhair...How is everything going ?
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@Ankit Kumar Jain – Its going ok. Whagt about you? Why you closed whatsapp?
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@Md Zuhair – Just like that...I will install it soon.
How did you solve the problem ? Do you know a solution using classical inequalities?
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@Ankit Kumar Jain – No . i solved it using calculus
f ( x ) = x 2 + 2 x 2 + 4 x f ′ ( x ) = ( x 2 + 2 ) 2 4 x + 8 − 4 x 2
Setting f ′ ( x ) = 0 we find x = 2 , − 1 as the critical points and further checking gives f ′ ′ ( 2 ) < 0 , f ′ ′ ( − 1 ) > 0
So M = f ( 2 ) = 2 , m = f ( − 1 ) = − 1 making the answer M − m = 3
Its overrated! Isnt it @Aditya Narayan Sharma
Ya. Maybe because people at a glance are considering the expression hard to bound using classical inequalities which is always a first choice
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Hmm, i always prefer one variable equations with calculus,
f ( x ) = 1 + x 2 + 2 4 x − 2 . ∴ f ′ ( x ) = ( x 2 + 2 ) 2 4 ( x 2 + 2 ) − 2 x ( 4 x − 2 ) = 0 . ∴ x = − 1 , 2 a s t h e c r i t i c a l p o i n t s . . . . . . J u s t a v a r i a t i o n .
x 2 + 2 x 2 + 4 x = y
⇒ x 2 ( y − 1 ) − 4 x + 2 y = 0
⇒ D = 1 6 − 8 y ( y − 1 ) ≥ 0
⇒ y ∈ [ − 1 , 2 ]
∴ M = 2 , m = − 1