sometimes inverties are cool
f 3 ( x ) − ( x 3 + 2 ) f 2 ( x ) + ( 2 x 3 + 1 ) f ( x ) − x 3 = 0
Function f is invertible from R → R and satisfies the equation above. Find f ′ ( 8 ) × ( f − 1 ) ′ ( 8 ) .
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1 solution
But sir, isn't ( f − 1 ( x ) ) ′ = 3 x 2 / 3 1 ? So you'll get ( f − 1 ( 8 ) ) ′ = 1 2 1 .
The equation factors as ( f ( x ) − 1 ) 2 ( f ( x ) − x 3 ) = 0 , so it's not hard to see that f ( x ) = x 3 since f is invertible. Then f ′ ( 8 ) = 1 9 2 and ( f − 1 ) ′ ( 8 ) = 1 / 1 2 , so the answer is 1 6 .