Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.

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Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is 12. Explanation: Given that y = –x3 + 3x2 + 9x – 27 "dy"/'dx" = – 3x2 + 6x + 9 ∴ Slope of the given curve, m = – 3x2 + 6x + 9 ....("dy"/"dx" = "m") "dm"/"dx" = –6x + 6 For local maxima and local minima, "dm"/"dx" = 0 ∴ – 6x + 6 = 0 ⇒ x = 1 Now ("d"^2"m")/("dx"^2) = = – 6 < 0 maxima ∴ Maximum value of the slope at x = 1 is "m"_(x = 1) = – 3(1)2 + 6(1) + 9 = – 3 + 6 + 9 = 12

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