A calculus problem by Shashwat Avasthi
If for all real , then find independent of and . Submit your answer as .
The answer is 16.
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f ( x ) = x 3 + x 2 f ′ ( 1 ) + x f ′ ′ ( 2 ) + f ′ ′ ′ ( 3 ) ⇒ f ′ ′ ′ ( 3 ) = 6 ; f ′ ′ ( 2 ) = 1 2 + 2 f ′ ( 1 ) ; f ′ ( 1 ) = 3 + 2 f ′ ( 1 ) + f ′ ′ ( 2 ) ⇒ f ′ ′ ( 2 ) = − f ′ ( 1 ) − 3 = 1 2 + 2 f ′ ( 1 ) ⇒ f ′ ( 1 ) = − 5 ⇒ f ′ ′ ( 2 ) = 2 ⇒ f ( 5 ) = 1 2 5 − 1 2 5 + 1 0 + 6 = 1 6