Let f ( x ) = x 5 + 2 x 3 + 3 x + 1 , x ∈ R , and g ( x ) be a function such that g ( f ( x )) = x for all x ∈ R . Then g ′ ( 7 ) g ( 7 ) is equal to :

Question

Let f ( x ) = x 5 + 2 x 3 + 3 x + 1 , x ∈ R , and g ( x ) be a function such that g ( f ( x )) = x for all x ∈ R . Then g ′ ( 7 ) g ( 7 ) ​ is equal to :

TestHub · Accepted Answer

​ f ( x ) = x 5 + 2 x 3 + 3 x + 1 f ′ ( x ) = 5 x 4 + 6 x 2 + 3 f ′ ( 1 ) = 5 + 6 + 3 = 14 g ( f ( x )) = x g ′ ( f ( x )) f ′ ( x ) = 1 for f ( x ) = 7 ⇒ x 5 + 2 x 3 + 3 x + 1 = 7 ⇒ x = 1 g ′ ( 7 ) f ′ ( 1 ) = 1 ⇒ g ′ ( 7 ) = f ′ ( 1 ) 1 ​ = 14 1 ​ ​ ​ x = 1 , f ( x ) = 7 ⇒ g ( 7 ) = 1 g ′ ( 7 ) g ( 7 ) ​ = 1/14 1 ​ = 14 ​

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