The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.

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The function f(x) = 2x3 – 3x2 – 12x + 4, has one maxima and one minima. Explanation: We have f(x) = 2x3 – 3x2 – 12x + 4 f'(x) = 6x2 – 6x – 12 For local maxima and local minima f'(x) = 0 ∴ 6x2 – 6x – 12 = 0 ⇒ x2 – x – 2 = 0 ⇒ x2 – 2x + x – 2 = 0 ⇒ x(x – 2) + 1(x – 2) = 0 ⇒ (x + 1)(x – 2) = 0 x = –1, 2 are the points of local maxima and local minima Now f'(x) = 12x – 6 "f''"(x)_(x = -1) = 12(–1) – 6 = – 12 – 6 = – 18 < 0, maxima "f''"(x)_(x = 2) = 12(2) – 6 = 24 – 6 = 18 > 0 minima So, the function is maximum at x = –1 and minimum at x = 2

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