Let g ( x ) = 3 f x 3 + f ( 3 - x ) and f " ( x ) 0 for all x ∈ ( 0 , 3 ) . If g is decreasing in ( 0 , α ) and increasing in ( α , 3 ) , then 8 α is

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Let g ( x ) = 3 f x 3 + f ( 3 - x ) and f " ( x ) > 0 for all x ∈ ( 0 , 3 ) . If g is decreasing in ( 0 , α ) and increasing in ( α , 3 ) , then 8 α is

Quantrex Academy · Accepted Answer

Correct answer: C. 18

Given: g ( x ) = 3 f x 3 + f ( 3 - x ) and f " ( x ) > 0 ∀ x ∈ ( 0 , 3 ) ⇒ f ' ( x ) is increasing function ⇒ g ' ( x ) = 3 × 1 3 · f ' x 3 - f ' ( 3 - x ) ⇒ g ' ( x ) = f ' x 3 - f ' ( 3 - x ) It is given that g is decreasing in 0 , α then g ' ( x ) < 0 . ⇒ f ' x 3 - f ' ( 3 - x ) < 0 ⇒ f ' x 3 < f ' ( 3 - x ) ⇒ x 3 < 3 - x ⇒ x < 9 - 3 x ⇒ x < 9 4 ⇒ α = 9 4 ⇒ 8 α = 8 × 9 4 ⇒ 8 α = 18

Let g ( x ) = 3 f x 3 + f ( 3 - x ) and f " ( x ) > 0 for all x ∈ ( 0 , 3 ) . If g is decreasing in ( 0 , α ) and increasing in ( α , 3 ) , then 8 α is

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