Free Mathematical Methods resources · VCE Units 3 & 4
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Let f(x) = 2x^3 - 9x^2 + 12x. Find the coordinates of the stationary points of f and state the nature of each.
f'(x) = 6x^2 - 18x + 12 (1 mark for correct derivative). Set f'(x) = 0: 6(x^2 - 3x + 2) = 0, so (x - 1)(x - 2) = 0, giving x = 1 and x = 2 (1 mark for both x-values). f(1) = 2 - 9 + 12 = 5 and f(2) = 16 - 36 + 24 = 4, so stationary points are (1, 5) and (2, 4) (1 mark for both coordinates). Testing the sign of f'(x) either side (or noting the positive leading coefficient of a cubic), (1, 5) is a local maximum and (2, 4) is a local minimum (1 mark for correct nature of both). Total 4 marks.
Evaluate the definite integral of (3x^2 - 4x + 1) with respect to x, from x = 0 to x = 2.
Antiderivative is x^3 - 2x^2 + x (1 mark). At x = 2: 8 - 8 + 2 = 2; at x = 0: 0 (1 mark for substituting both terminals). Value = 2 - 0 = 2 (1 mark). Total 3 marks.
More to practise: all Mathematical Methods practice questions.