If f ( x ) = 3 x 10 − 7 x 8 + 5 x 6 − 21 x 3 + 3 x 2 − 7 , then lim α → 0 α 3 + 3 α f ( 1 − α ) − f ( 1 ) is

Question

If f ( x ) = 3 x 10 − 7 x 8 + 5 x 6 − 21 x 3 + 3 x 2 − 7 , then lim α → 0 ​ α 3 + 3 α f ( 1 − α ) − f ( 1 ) ​ is

TestHub · Accepted Answer

Let f ′ ( x ) f ′ ( x ) f ′ ( 1 ) ​ = 3 x 10 − 7 x 8 + 5 x 6 − 21 x 3 + 3 x 2 − 7 = 30 x 9 − 56 x 7 + 30 x 5 − 63 x 2 + 6 x = 30 − 56 + 30 − 63 + 6 = 66 − 63 − 56 = − 53 ​ ​ Consider α → 0 lim ​ α 3 + 3 α f ( 1 − α ) − f ( 1 ) ​ = α → 0 lim ​ 3 α 2 + 3 f ′ ( 1 − α ) ( − 1 ) − 0 ​ (By using ​ L'hospital rule) = 3 ( 0 ) 2 + 3 f ′ ( 1 − 0 ) ( − 1 ) ​ = 3 − f ′ ( 1 ) ​ = 3 53 ​

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