4PM1/02 Jun 2019, Question 9: Stationary Points and Turning Points

Part (a) · 4 marks

Integrate f′(x), use the given point to find the constant, and show the curve passes through the origin.

1. 1f(x) = x⁴/4 − x³/3 − 2x² + 4x + c
2. 2At (−2, −28/3): −28/3 = 4 + 8/3 − 8 − 8 + c
3. 3c = 0
4. 4f(0) = 0, so C passes through the origin

Answerf(x) = x⁴/4 − x³/3 − 2x² + 4x (c = 0), so f(0) = 0 and C passes through O

Part (b) · 7 marks

Show stationary points occur at x = 2 and x = 1, classify them, and find the exact y values there.

1. 1f′(2) = 8 − 4 − 8 + 4 = 0 and f′(1) = 1 − 1 − 4 + 4 = 0
2. 2f″(x) = 3x² − 2x − 4
3. 3f″(2) = 12 − 4 − 4 = 4 > 0: minimum at x = 2
4. 4f″(1) = 3 − 2 − 4 = −3 < 0: maximum at x = 1
5. 5f(1) = 1/4 − 1/3 − 2 + 4 = 23/12
6. 6f(2) = 4 − 8/3 − 8 + 8 = 4/3

Answer(i) f′(2) = f′(1) = 0; f″(2) = 4 > 0 so minimum at x = 2, f″(1) = −3 < 0 so maximum at x = 1 (ii) at x = 1, y = 23/12 (1 11/12); at x = 2, y = 4/3 (1 1/3)

Part (c) · 3 marks

Find the coordinates of the remaining turning point A and determine whether it is a maximum or minimum.

1. 1f′(x) = (x − 1)(x − 2)(x + 2), so the third turning point is at x = −2
2. 2y = −28/3 (the given point), A(−2, −28/3)
3. 3f″(−2) = 12 + 4 − 4 = 12 > 0, so A is a minimum

Answer(i) A(−2, −28/3) (ii) minimum

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