Maxima and minima

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Question

$Maximum slope of the curve y = - x^{3} + 3 x^{2} + 9 x - 27 is$

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Solution

$\begin{array}{l} Slope = \frac{d y}{d x} = m = - 3 x^{2} + 6 x + 9 \\ \frac{d m}{d x} = - 6 x + 6 \\ \frac{d m}{d x} = 0 \Rightarrow x = 1 \\ \frac{d^{2} m}{d x^{2}} = - 6 < 0, \forall x \\ x = 1 is point of maximum \\ Max. slope = - 3 (1)^{2} + 6 (1) + 9 = 12 \end{array}$

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