The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.

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The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are (2, 2), (–2, 34). Explanation: Given that y = x3 – 12x + 18 Differentiating both sides w.r.t. x, we have $\Rightarrow$ "dy"/"dx" = 3x2 – 12 Since the tangents are parallel to x-axis, then "dy"/"dx" = 0 $\therefore$ 3x2 – 12 = 0 $\Rightarrow$ x = $\pm$ 2 $\therefore$ y_(x = 2) = (2)3 – 12(2) + 18 = 8 – 24 + 18 = 2 y_(x = -2) = (– 2)3 – 12 (– 2) + 18 = – 8 + 24 + 18 = 34 $\therefore$ Points are (2, 2) and (– 2, 34).

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