Let f be a cubic polynomial with a local maximum at x = 2 . Also, f / has a local minimum at x = 4 , and f ( 2 ) = 12 , f ( 4 ) = 4 . Find the minimum value of f on [1, 6].

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Writing f ′′ = λ ( x − 4 ) and using f ′ ( 2 ) = 0 , f ( 2 ) = 12 , f ( 4 ) = 4 gives f ( x ) = x 3 /2 − 6 x 2 + 18 x − 4 . Its critical and endpoint values on [1, 6] show the minimum is f ( 6 ) = − 4 .

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